Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 73
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.31e11·2-s + 7.82e15·3-s + 7.94e21·4-s − 7.75e24·5-s + 1.03e27·6-s + 1.05e30·7-s − 1.97e32·8-s − 6.75e34·9-s − 1.02e36·10-s + 5.33e37·11-s + 6.22e37·12-s − 2.33e40·13-s + 1.39e41·14-s − 6.07e40·15-s − 1.01e44·16-s − 3.97e44·17-s − 8.90e45·18-s − 7.99e46·19-s − 6.16e46·20-s + 8.29e45·21-s + 7.04e48·22-s + 6.65e49·23-s − 1.54e48·24-s − 9.98e50·25-s − 3.08e51·26-s − 1.05e51·27-s + 8.42e51·28-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.0301·3-s + 0.841·4-s − 0.238·5-s + 0.0408·6-s + 0.151·7-s − 0.215·8-s − 0.999·9-s − 0.323·10-s + 0.520·11-s + 0.0253·12-s − 0.512·13-s + 0.204·14-s − 0.00717·15-s − 1.13·16-s − 0.487·17-s − 1.35·18-s − 1.69·19-s − 0.200·20-s + 0.00454·21-s + 0.706·22-s + 1.31·23-s − 0.00647·24-s − 0.943·25-s − 0.695·26-s − 0.0602·27-s + 0.127·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(73\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :73/2),\ -1)\)
\(L(37)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{75}{2})\)   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 \( 1 - 1.31e11T + 9.44e21T^{2} \)
3 \( 1 - 7.82e15T + 6.75e34T^{2} \)
5 \( 1 + 7.75e24T + 1.05e51T^{2} \)
7 \( 1 - 1.05e30T + 4.92e61T^{2} \)
11 \( 1 - 5.33e37T + 1.05e76T^{2} \)
13 \( 1 + 2.33e40T + 2.07e81T^{2} \)
17 \( 1 + 3.97e44T + 6.64e89T^{2} \)
19 \( 1 + 7.99e46T + 2.23e93T^{2} \)
23 \( 1 - 6.65e49T + 2.54e99T^{2} \)
29 \( 1 - 2.50e53T + 5.68e106T^{2} \)
31 \( 1 + 3.03e54T + 7.40e108T^{2} \)
37 \( 1 + 2.59e57T + 3.01e114T^{2} \)
41 \( 1 - 1.11e58T + 5.41e117T^{2} \)
43 \( 1 + 1.82e58T + 1.75e119T^{2} \)
47 \( 1 - 1.00e61T + 1.15e122T^{2} \)
53 \( 1 - 9.84e60T + 7.44e125T^{2} \)
59 \( 1 - 5.71e64T + 1.87e129T^{2} \)
61 \( 1 + 7.84e64T + 2.13e130T^{2} \)
67 \( 1 - 5.03e66T + 2.01e133T^{2} \)
71 \( 1 - 4.24e67T + 1.38e135T^{2} \)
73 \( 1 - 9.03e67T + 1.05e136T^{2} \)
79 \( 1 - 2.30e69T + 3.36e138T^{2} \)
83 \( 1 + 1.28e70T + 1.23e140T^{2} \)
89 \( 1 + 2.21e71T + 2.02e142T^{2} \)
97 \( 1 + 2.42e72T + 1.08e145T^{2} \)
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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

−15.19419804383205524147371171677, −14.11042002127587830761895302861, −12.56913744951565258222083984394, −11.22415294436148200771855923588, −8.763827905758262063495101613042, −6.58119676321529783002704159117, −5.12999744947285585775759446480, −3.79574254300370297129996634068, −2.37016929090892543548118267949, 0, 2.37016929090892543548118267949, 3.79574254300370297129996634068, 5.12999744947285585775759446480, 6.58119676321529783002704159117, 8.763827905758262063495101613042, 11.22415294436148200771855923588, 12.56913744951565258222083984394, 14.11042002127587830761895302861, 15.19419804383205524147371171677

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