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  − 3.23e8·2-s − 8.24e12·3-s − 4.71e17·4-s − 7.71e20·5-s + 2.67e21·6-s − 4.97e24·7-s + 3.39e26·8-s − 1.40e28·9-s + 2.49e29·10-s − 5.41e30·11-s + 3.89e30·12-s − 5.16e32·13-s + 1.61e33·14-s + 6.36e33·15-s + 1.62e35·16-s − 9.94e35·17-s + 4.55e36·18-s − 3.90e37·19-s + 3.63e38·20-s + 4.10e37·21-s + 1.75e39·22-s + 2.19e40·23-s − 2.79e39·24-s + 4.22e41·25-s + 1.67e41·26-s + 2.32e41·27-s + 2.34e42·28-s + ⋯
L(s)  = 1  − 0.426·2-s − 0.0694·3-s − 0.818·4-s − 1.85·5-s + 0.0295·6-s − 0.584·7-s + 0.775·8-s − 0.995·9-s + 0.790·10-s − 1.02·11-s + 0.0567·12-s − 0.710·13-s + 0.249·14-s + 0.128·15-s + 0.487·16-s − 0.500·17-s + 0.424·18-s − 0.738·19-s + 1.51·20-s + 0.0405·21-s + 0.438·22-s + 1.48·23-s − 0.0538·24-s + 2.43·25-s + 0.303·26-s + 0.138·27-s + 0.478·28-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  =  $(2,\ 1,\ (\ :59/2),\ 1)$
$L(30)$  $\approx$  $0.0286742$
$L(\frac12)$  $\approx$  $0.0286742$
$L(\frac{61}{2})$   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 \( 1 + 3.23e8T + 5.76e17T^{2} \)
3 \( 1 + 8.24e12T + 1.41e28T^{2} \)
5 \( 1 + 7.71e20T + 1.73e41T^{2} \)
7 \( 1 + 4.97e24T + 7.25e49T^{2} \)
11 \( 1 + 5.41e30T + 2.76e61T^{2} \)
13 \( 1 + 5.16e32T + 5.28e65T^{2} \)
17 \( 1 + 9.94e35T + 3.94e72T^{2} \)
19 \( 1 + 3.90e37T + 2.79e75T^{2} \)
23 \( 1 - 2.19e40T + 2.19e80T^{2} \)
29 \( 1 + 1.47e43T + 1.91e86T^{2} \)
31 \( 1 + 9.00e43T + 9.78e87T^{2} \)
37 \( 1 + 4.57e45T + 3.34e92T^{2} \)
41 \( 1 - 1.81e46T + 1.42e95T^{2} \)
43 \( 1 + 1.81e48T + 2.36e96T^{2} \)
47 \( 1 - 9.84e48T + 4.50e98T^{2} \)
53 \( 1 + 4.04e50T + 5.39e101T^{2} \)
59 \( 1 + 2.73e52T + 3.02e104T^{2} \)
61 \( 1 + 2.82e52T + 2.16e105T^{2} \)
67 \( 1 + 7.96e53T + 5.47e107T^{2} \)
71 \( 1 + 2.85e54T + 1.67e109T^{2} \)
73 \( 1 - 1.64e55T + 8.63e109T^{2} \)
79 \( 1 + 1.63e56T + 9.12e111T^{2} \)
83 \( 1 + 4.71e55T + 1.68e113T^{2} \)
89 \( 1 - 1.35e57T + 1.03e115T^{2} \)
97 \( 1 - 1.09e58T + 1.65e117T^{2} \)
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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

−18.80221000943775983000737836644, −16.76996522828622469219984359819, −15.06204475417501662870263730955, −12.78343589796444345725042089059, −10.97690379856022978007887902437, −8.802134135045037918398201960027, −7.53689042800403740235672257470, −4.82438506927423381864202585868, −3.25519075822932993127530371829, −0.11548805786913623004689854185, 0.11548805786913623004689854185, 3.25519075822932993127530371829, 4.82438506927423381864202585868, 7.53689042800403740235672257470, 8.802134135045037918398201960027, 10.97690379856022978007887902437, 12.78343589796444345725042089059, 15.06204475417501662870263730955, 16.76996522828622469219984359819, 18.80221000943775983000737836644

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