Properties

Degree 2
Conductor $ 2 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 8-s − 2·9-s − 3·10-s + 12-s + 5·13-s + 14-s + 3·15-s + 16-s + 2·18-s + 2·19-s + 3·20-s − 21-s − 6·23-s − 24-s + 4·25-s − 5·26-s − 5·27-s − 28-s − 3·30-s − 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.458·19-s + 0.670·20-s − 0.218·21-s − 1.25·23-s − 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s − 0.547·30-s − 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(158\)    =    \(2 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{158} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 158,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.123529159$
$L(\frac12)$  $\approx$  $1.123529159$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
79 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.21072161753896, −18.20898460118204, −17.72784636407049, −16.68513339409130, −15.93051252033853, −14.70228087364002, −13.80067442652171, −13.20723561594025, −11.75648260207424, −10.63047286146988, −9.683357631351871, −8.946470375618335, −8.008923035003699, −6.436986195998011, −5.666748622859454, −3.354036950212796, −1.901472845664488, 1.901472845664488, 3.354036950212796, 5.666748622859454, 6.436986195998011, 8.008923035003699, 8.946470375618335, 9.683357631351871, 10.63047286146988, 11.75648260207424, 13.20723561594025, 13.80067442652171, 14.70228087364002, 15.93051252033853, 16.68513339409130, 17.72784636407049, 18.20898460118204, 19.21072161753896

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