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Properties

Label 4-189e2-1.1-c0e2-0-0
Degree $4$
Conductor $35721$
Sign $1$
Analytic cond. $0.00889688$
Root an. cond. $0.307120$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·7-s − 2·13-s − 2·19-s − 25-s − 2·28-s + 31-s + 37-s − 2·43-s + 3·49-s + 2·52-s + 61-s + 64-s + 67-s − 2·73-s + 2·76-s + 79-s − 4·91-s − 2·97-s + 100-s + 103-s + 109-s − 121-s − 124-s + 127-s + 131-s − 4·133-s + ⋯
L(s)  = 1  − 4-s + 2·7-s − 2·13-s − 2·19-s − 25-s − 2·28-s + 31-s + 37-s − 2·43-s + 3·49-s + 2·52-s + 61-s + 64-s + 67-s − 2·73-s + 2·76-s + 79-s − 4·91-s − 2·97-s + 100-s + 103-s + 109-s − 121-s − 124-s + 127-s + 131-s − 4·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35721\)    =    \(3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00889688\)
Root analytic conductor: \(0.307120\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 35721,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4252050236\)
\(L(\frac12)\) \(\approx\) \(0.4252050236\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.95712025298803857390028926301, −12.69411025483152826804369717737, −11.84183935124613510867712083098, −11.76185879594565457888328419177, −11.18394374517780049218032169032, −10.56154615413713456608029130908, −9.962045875563006616284826861808, −9.771200136271633034970017040564, −8.752839438830055882330236675064, −8.703775877156780684159818138802, −7.88639050962153812660136432658, −7.80499230711910464163221183475, −6.92610513305897376346028673064, −6.30087174016502298916855520295, −5.19214167103579096857904269941, −5.11203554069154677044449361215, −4.26297223858260436199025984889, −4.20744014396295569953301800008, −2.55939948811483532521196998680, −1.91255076604218535700168097614, 1.91255076604218535700168097614, 2.55939948811483532521196998680, 4.20744014396295569953301800008, 4.26297223858260436199025984889, 5.11203554069154677044449361215, 5.19214167103579096857904269941, 6.30087174016502298916855520295, 6.92610513305897376346028673064, 7.80499230711910464163221183475, 7.88639050962153812660136432658, 8.703775877156780684159818138802, 8.752839438830055882330236675064, 9.771200136271633034970017040564, 9.962045875563006616284826861808, 10.56154615413713456608029130908, 11.18394374517780049218032169032, 11.76185879594565457888328419177, 11.84183935124613510867712083098, 12.69411025483152826804369717737, 12.95712025298803857390028926301

Graph of the $Z$-function along the critical line