Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 59
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.92e8·2-s − 1.03e14·3-s − 9.64e16·4-s + 3.87e20·5-s − 7.15e22·6-s − 7.47e23·7-s − 4.66e26·8-s − 3.46e27·9-s + 2.68e29·10-s + 4.64e30·11-s + 9.95e30·12-s + 1.02e33·13-s − 5.17e32·14-s − 3.99e34·15-s − 2.67e35·16-s + 2.49e36·17-s − 2.39e36·18-s + 3.24e37·19-s − 3.73e37·20-s + 7.71e37·21-s + 3.22e39·22-s + 2.77e40·23-s + 4.81e40·24-s − 2.36e40·25-s + 7.13e41·26-s + 1.81e42·27-s + 7.20e40·28-s + ⋯
L(s)  = 1  + 0.912·2-s − 0.868·3-s − 0.167·4-s + 0.929·5-s − 0.792·6-s − 0.0877·7-s − 1.06·8-s − 0.245·9-s + 0.847·10-s + 0.883·11-s + 0.145·12-s + 1.41·13-s − 0.0800·14-s − 0.807·15-s − 0.804·16-s + 1.25·17-s − 0.223·18-s + 0.613·19-s − 0.155·20-s + 0.0762·21-s + 0.806·22-s + 1.87·23-s + 0.925·24-s − 0.136·25-s + 1.29·26-s + 1.08·27-s + 0.0146·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(60-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(59\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :59/2),\ 1)$
$L(30)$  $\approx$  $2.41757$
$L(\frac12)$  $\approx$  $2.41757$
$L(\frac{61}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 - 6.92e8T + 5.76e17T^{2} \)
3 \( 1 + 1.03e14T + 1.41e28T^{2} \)
5 \( 1 - 3.87e20T + 1.73e41T^{2} \)
7 \( 1 + 7.47e23T + 7.25e49T^{2} \)
11 \( 1 - 4.64e30T + 2.76e61T^{2} \)
13 \( 1 - 1.02e33T + 5.28e65T^{2} \)
17 \( 1 - 2.49e36T + 3.94e72T^{2} \)
19 \( 1 - 3.24e37T + 2.79e75T^{2} \)
23 \( 1 - 2.77e40T + 2.19e80T^{2} \)
29 \( 1 + 1.27e43T + 1.91e86T^{2} \)
31 \( 1 + 1.14e44T + 9.78e87T^{2} \)
37 \( 1 - 2.85e46T + 3.34e92T^{2} \)
41 \( 1 + 2.41e47T + 1.42e95T^{2} \)
43 \( 1 - 1.26e48T + 2.36e96T^{2} \)
47 \( 1 + 1.54e49T + 4.50e98T^{2} \)
53 \( 1 - 3.50e50T + 5.39e101T^{2} \)
59 \( 1 - 3.68e51T + 3.02e104T^{2} \)
61 \( 1 + 9.17e51T + 2.16e105T^{2} \)
67 \( 1 - 4.08e53T + 5.47e107T^{2} \)
71 \( 1 + 3.75e52T + 1.67e109T^{2} \)
73 \( 1 + 2.19e54T + 8.63e109T^{2} \)
79 \( 1 - 1.38e56T + 9.12e111T^{2} \)
83 \( 1 - 1.05e56T + 1.68e113T^{2} \)
89 \( 1 + 3.89e57T + 1.03e115T^{2} \)
97 \( 1 - 7.21e58T + 1.65e117T^{2} \)
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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

−18.24475367595760616783208269753, −16.81142260218467368886178250877, −14.44500257566445186670671679281, −13.06227460911484865533818737305, −11.38617072594748556508167778935, −9.245452262231743604694207948846, −6.19121117951191906941621640811, −5.32663091084914999311716507641, −3.42616382507351596843680429771, −1.03375234070354278860348613847, 1.03375234070354278860348613847, 3.42616382507351596843680429771, 5.32663091084914999311716507641, 6.19121117951191906941621640811, 9.245452262231743604694207948846, 11.38617072594748556508167778935, 13.06227460911484865533818737305, 14.44500257566445186670671679281, 16.81142260218467368886178250877, 18.24475367595760616783208269753

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