Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $37$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.86e5·2-s − 2.05e7·3-s − 5.56e10·4-s − 8.29e12·5-s − 5.89e12·6-s + 1.97e15·7-s − 5.52e16·8-s − 4.49e17·9-s − 2.37e18·10-s − 2.57e19·11-s + 1.14e18·12-s + 5.42e20·13-s + 5.64e20·14-s + 1.70e20·15-s − 8.14e21·16-s − 3.52e22·17-s − 1.28e23·18-s + 6.82e23·19-s + 4.61e23·20-s − 4.06e22·21-s − 7.35e24·22-s − 8.19e22·23-s + 1.13e24·24-s − 3.91e24·25-s + 1.55e26·26-s + 1.85e25·27-s − 1.09e26·28-s + ⋯
L(s)  = 1  + 0.771·2-s − 0.0306·3-s − 0.404·4-s − 0.972·5-s − 0.0236·6-s + 0.458·7-s − 1.08·8-s − 0.999·9-s − 0.750·10-s − 1.39·11-s + 0.0124·12-s + 1.33·13-s + 0.353·14-s + 0.0298·15-s − 0.431·16-s − 0.608·17-s − 0.770·18-s + 1.50·19-s + 0.393·20-s − 0.0140·21-s − 1.07·22-s − 0.00526·23-s + 0.0332·24-s − 0.0538·25-s + 1.03·26-s + 0.0613·27-s − 0.185·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(38-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+37/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(37\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :37/2),\ -1)\)

Particular Values

\(L(19)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.86e5T + 1.37e11T^{2} \)
3 \( 1 + 2.05e7T + 4.50e17T^{2} \)
5 \( 1 + 8.29e12T + 7.27e25T^{2} \)
7 \( 1 - 1.97e15T + 1.85e31T^{2} \)
11 \( 1 + 2.57e19T + 3.40e38T^{2} \)
13 \( 1 - 5.42e20T + 1.64e41T^{2} \)
17 \( 1 + 3.52e22T + 3.36e45T^{2} \)
19 \( 1 - 6.82e23T + 2.06e47T^{2} \)
23 \( 1 + 8.19e22T + 2.42e50T^{2} \)
29 \( 1 + 1.51e27T + 1.28e54T^{2} \)
31 \( 1 - 2.60e27T + 1.51e55T^{2} \)
37 \( 1 + 1.30e29T + 1.05e58T^{2} \)
41 \( 1 + 4.07e29T + 4.70e59T^{2} \)
43 \( 1 + 2.92e30T + 2.74e60T^{2} \)
47 \( 1 - 3.58e30T + 7.37e61T^{2} \)
53 \( 1 - 3.56e31T + 6.28e63T^{2} \)
59 \( 1 + 3.03e32T + 3.32e65T^{2} \)
61 \( 1 - 1.16e33T + 1.14e66T^{2} \)
67 \( 1 + 2.44e33T + 3.67e67T^{2} \)
71 \( 1 - 6.30e33T + 3.13e68T^{2} \)
73 \( 1 - 1.02e34T + 8.76e68T^{2} \)
79 \( 1 - 1.20e35T + 1.63e70T^{2} \)
83 \( 1 + 3.26e35T + 1.01e71T^{2} \)
89 \( 1 + 1.56e36T + 1.34e72T^{2} \)
97 \( 1 + 1.07e36T + 3.24e73T^{2} \)
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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

−22.76929614383051956740114757090, −20.63653358861324988635101867697, −18.24871950005246888891702940735, −15.51415077536740871668288185592, −13.58579983358184777956746156554, −11.50132024290830732411905645660, −8.313372276419025684925283884069, −5.30469258083104533062022796873, −3.40942703835798794686937639842, 0, 3.40942703835798794686937639842, 5.30469258083104533062022796873, 8.313372276419025684925283884069, 11.50132024290830732411905645660, 13.58579983358184777956746156554, 15.51415077536740871668288185592, 18.24871950005246888891702940735, 20.63653358861324988635101867697, 22.76929614383051956740114757090

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