Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 93
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.20e12·2-s − 1.59e22·3-s − 9.87e27·4-s − 3.34e32·5-s + 8.30e34·6-s − 2.46e38·7-s + 1.02e41·8-s + 1.91e43·9-s + 1.74e45·10-s + 2.07e48·11-s + 1.57e50·12-s + 2.80e51·13-s + 1.28e51·14-s + 5.33e54·15-s + 9.72e55·16-s − 1.34e57·17-s − 9.96e55·18-s + 5.04e59·19-s + 3.30e60·20-s + 3.93e60·21-s − 1.08e61·22-s − 1.78e63·23-s − 1.64e63·24-s + 1.08e64·25-s − 1.45e64·26-s + 3.45e66·27-s + 2.43e66·28-s + ⋯
L(s)  = 1  − 0.0523·2-s − 1.03·3-s − 0.997·4-s − 1.05·5-s + 0.0543·6-s − 0.124·7-s + 0.104·8-s + 0.0812·9-s + 0.0550·10-s + 0.781·11-s + 1.03·12-s + 0.446·13-s + 0.00650·14-s + 1.09·15-s + 0.991·16-s − 0.820·17-s − 0.00424·18-s + 1.74·19-s + 1.04·20-s + 0.129·21-s − 0.0409·22-s − 0.855·23-s − 0.108·24-s + 0.107·25-s − 0.0233·26-s + 0.955·27-s + 0.124·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(94-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+93/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(93\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :93/2),\ -1)\)
\(L(47)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{95}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 5.20e12T + 9.90e27T^{2} \)
3 \( 1 + 1.59e22T + 2.35e44T^{2} \)
5 \( 1 + 3.34e32T + 1.00e65T^{2} \)
7 \( 1 + 2.46e38T + 3.92e78T^{2} \)
11 \( 1 - 2.07e48T + 7.07e96T^{2} \)
13 \( 1 - 2.80e51T + 3.95e103T^{2} \)
17 \( 1 + 1.34e57T + 2.70e114T^{2} \)
19 \( 1 - 5.04e59T + 8.39e118T^{2} \)
23 \( 1 + 1.78e63T + 4.37e126T^{2} \)
29 \( 1 + 1.40e68T + 1.00e136T^{2} \)
31 \( 1 - 3.47e69T + 4.97e138T^{2} \)
37 \( 1 - 3.66e72T + 6.96e145T^{2} \)
41 \( 1 - 5.70e74T + 9.74e149T^{2} \)
43 \( 1 - 1.54e75T + 8.17e151T^{2} \)
47 \( 1 - 7.65e77T + 3.19e155T^{2} \)
53 \( 1 + 2.23e80T + 2.27e160T^{2} \)
59 \( 1 + 3.20e82T + 4.88e164T^{2} \)
61 \( 1 + 1.20e83T + 1.08e166T^{2} \)
67 \( 1 - 6.65e84T + 6.68e169T^{2} \)
71 \( 1 - 8.45e85T + 1.46e172T^{2} \)
73 \( 1 - 7.12e86T + 1.94e173T^{2} \)
79 \( 1 + 1.51e87T + 3.01e176T^{2} \)
83 \( 1 - 1.58e89T + 2.98e178T^{2} \)
89 \( 1 + 1.03e90T + 1.96e181T^{2} \)
97 \( 1 - 1.83e92T + 5.88e184T^{2} \)
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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

−13.93021585873989148730532850219, −12.19698778617812494232706452581, −11.20063599103942584900754614849, −9.379596080325986618591900029371, −7.85737614058130826612856607251, −6.08732055577577117011239224737, −4.68893646141232147844535620846, −3.57669082731974987690155176724, −0.957896353390260094617286693473, 0, 0.957896353390260094617286693473, 3.57669082731974987690155176724, 4.68893646141232147844535620846, 6.08732055577577117011239224737, 7.85737614058130826612856607251, 9.379596080325986618591900029371, 11.20063599103942584900754614849, 12.19698778617812494232706452581, 13.93021585873989148730532850219

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