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  − 5.90e9·2-s + 3.95e17·3-s − 9.40e21·4-s + 1.69e25·5-s − 2.33e27·6-s − 2.69e30·7-s + 1.11e32·8-s + 8.86e34·9-s − 9.98e34·10-s − 1.46e38·11-s − 3.71e39·12-s − 3.34e40·13-s + 1.59e40·14-s + 6.68e42·15-s + 8.82e43·16-s − 1.35e45·17-s − 5.23e44·18-s + 7.35e46·19-s − 1.59e47·20-s − 1.06e48·21-s + 8.67e47·22-s + 2.07e49·23-s + 4.40e49·24-s − 7.72e50·25-s + 1.97e50·26-s + 8.32e51·27-s + 2.54e52·28-s + ⋯
L(s)  = 1  − 0.0607·2-s + 1.52·3-s − 0.996·4-s + 0.519·5-s − 0.0924·6-s − 0.384·7-s + 0.121·8-s + 1.31·9-s − 0.0315·10-s − 1.43·11-s − 1.51·12-s − 0.734·13-s + 0.0233·14-s + 0.789·15-s + 0.988·16-s − 1.66·17-s − 0.0797·18-s + 1.55·19-s − 0.517·20-s − 0.585·21-s + 0.0870·22-s + 0.411·23-s + 0.184·24-s − 0.730·25-s + 0.0446·26-s + 0.473·27-s + 0.383·28-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 \( 1 + 5.90e9T + 9.44e21T^{2} \)
3 \( 1 - 3.95e17T + 6.75e34T^{2} \)
5 \( 1 - 1.69e25T + 1.05e51T^{2} \)
7 \( 1 + 2.69e30T + 4.92e61T^{2} \)
11 \( 1 + 1.46e38T + 1.05e76T^{2} \)
13 \( 1 + 3.34e40T + 2.07e81T^{2} \)
17 \( 1 + 1.35e45T + 6.64e89T^{2} \)
19 \( 1 - 7.35e46T + 2.23e93T^{2} \)
23 \( 1 - 2.07e49T + 2.54e99T^{2} \)
29 \( 1 + 3.30e53T + 5.68e106T^{2} \)
31 \( 1 - 1.78e54T + 7.40e108T^{2} \)
37 \( 1 + 3.22e57T + 3.01e114T^{2} \)
41 \( 1 + 6.12e58T + 5.41e117T^{2} \)
43 \( 1 - 2.07e59T + 1.75e119T^{2} \)
47 \( 1 - 5.09e58T + 1.15e122T^{2} \)
53 \( 1 + 1.65e62T + 7.44e125T^{2} \)
59 \( 1 - 1.57e64T + 1.87e129T^{2} \)
61 \( 1 - 3.70e64T + 2.13e130T^{2} \)
67 \( 1 + 2.21e66T + 2.01e133T^{2} \)
71 \( 1 + 1.08e67T + 1.38e135T^{2} \)
73 \( 1 - 2.15e67T + 1.05e136T^{2} \)
79 \( 1 - 1.93e69T + 3.36e138T^{2} \)
83 \( 1 - 1.17e70T + 1.23e140T^{2} \)
89 \( 1 - 1.65e71T + 2.02e142T^{2} \)
97 \( 1 + 9.74e71T + 1.08e145T^{2} \)
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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

−15.42353542174442733641138718637, −13.82872871116858581075635973149, −13.13606163535545298771749329200, −9.945632686221817311334415610936, −8.962355286761724857259983449662, −7.60589516847797714562159159172, −5.06991403691409709398322010035, −3.36258067312358559770568212185, −2.11654228559532010207876022589, 0, 2.11654228559532010207876022589, 3.36258067312358559770568212185, 5.06991403691409709398322010035, 7.60589516847797714562159159172, 8.962355286761724857259983449662, 9.945632686221817311334415610936, 13.13606163535545298771749329200, 13.82872871116858581075635973149, 15.42353542174442733641138718637

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