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Properties

Label 2-208-52.51-c0-0-0
Degree $2$
Conductor $208$
Sign $1$
Analytic cond. $0.103805$
Root an. cond. $0.322188$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 13-s − 2·17-s + 25-s − 2·29-s − 49-s + 2·53-s + 2·61-s + 81-s + 2·101-s − 2·113-s − 117-s + ⋯
L(s)  = 1  + 9-s − 13-s − 2·17-s + 25-s − 2·29-s − 49-s + 2·53-s + 2·61-s + 81-s + 2·101-s − 2·113-s − 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.103805\)
Root analytic conductor: \(0.322188\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{208} (207, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7040449453\)
\(L(\frac12)\) \(\approx\) \(0.7040449453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( ( 1 + T )^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 + T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.84216826972720396534944438557, −11.62074117916663017153709902749, −10.69181790496113516813530649640, −9.676660089512915951413231397337, −8.782672525949915380726097618143, −7.40350736653843378830348360900, −6.67707391086418863494569206827, −5.10538667487405721881327138549, −4.04535655824561171760116937879, −2.20467624493930931798195661261, 2.20467624493930931798195661261, 4.04535655824561171760116937879, 5.10538667487405721881327138549, 6.67707391086418863494569206827, 7.40350736653843378830348360900, 8.782672525949915380726097618143, 9.676660089512915951413231397337, 10.69181790496113516813530649640, 11.62074117916663017153709902749, 12.84216826972720396534944438557

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