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Properties

Label 4-189e2-1.1-c0e2-0-1
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 − 25-s − 2·28-s − 3·31-s − 37-s + 2·43-s + 3·49-s + 3·61-s − 64-s + 67-s + 79-s − 100-s − 3·103-s − 109-s + 121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4-s − 2·7-s − 25-s − 2·28-s − 3·31-s − 37-s + 2·43-s + 3·49-s + 3·61-s − 64-s + 67-s + 79-s − 100-s − 3·103-s − 109-s + 121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4803776810\)
\(L(\frac12)\) \(\approx\) \(0.4803776810\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2^2$ \( 1 - T^{2} + T^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−12.74477142285516605760857436266, −12.73239221954578803744320670572, −12.14654922780244633590250917269, −11.61762879820832239756966354126, −10.87364776718320672025700466999, −10.85521901287101135048632228002, −10.03256821860987001826954254153, −9.626068553937824179479037158025, −9.165654541498292184752169336727, −8.729187256536063775811934105183, −7.75267689373509500079668521396, −7.31006683125025324870153340290, −6.75727971220309298784070821063, −6.51542422563339517944929414863, −5.59099571872574271090089067630, −5.48776915957869355677584670108, −3.88563800900704074452408537378, −3.75222531661024383459138765557, −2.74706031030906967687092263124, −2.08500484518962646615119309951, 2.08500484518962646615119309951, 2.74706031030906967687092263124, 3.75222531661024383459138765557, 3.88563800900704074452408537378, 5.48776915957869355677584670108, 5.59099571872574271090089067630, 6.51542422563339517944929414863, 6.75727971220309298784070821063, 7.31006683125025324870153340290, 7.75267689373509500079668521396, 8.729187256536063775811934105183, 9.165654541498292184752169336727, 9.626068553937824179479037158025, 10.03256821860987001826954254153, 10.85521901287101135048632228002, 10.87364776718320672025700466999, 11.61762879820832239756966354126, 12.14654922780244633590250917269, 12.73239221954578803744320670572, 12.74477142285516605760857436266

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