Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 69
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.12e10·2-s + 3.67e16·3-s − 1.38e20·4-s − 2.01e24·5-s + 7.80e26·6-s + 2.05e29·7-s − 1.54e31·8-s + 5.14e32·9-s − 4.28e34·10-s − 1.09e36·11-s − 5.08e36·12-s − 2.46e37·13-s + 4.37e39·14-s − 7.40e40·15-s − 2.47e41·16-s − 2.55e42·17-s + 1.09e43·18-s − 1.09e44·19-s + 2.79e44·20-s + 7.55e45·21-s − 2.31e46·22-s − 8.34e45·23-s − 5.68e47·24-s + 2.37e48·25-s − 5.24e47·26-s − 1.17e49·27-s − 2.85e49·28-s + ⋯
L(s)  = 1  + 0.874·2-s + 1.27·3-s − 0.234·4-s − 1.54·5-s + 1.11·6-s + 1.43·7-s − 1.08·8-s + 0.616·9-s − 1.35·10-s − 1.28·11-s − 0.298·12-s − 0.0914·13-s + 1.25·14-s − 1.96·15-s − 0.710·16-s − 0.903·17-s + 0.539·18-s − 0.837·19-s + 0.363·20-s + 1.82·21-s − 1.12·22-s − 0.0874·23-s − 1.37·24-s + 1.39·25-s − 0.0799·26-s − 0.487·27-s − 0.337·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\,\Lambda(70-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(69\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1,\ (\ :69/2),\ -1)$
$L(35)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{71}{2})$   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 \( 1 - 2.12e10T + 5.90e20T^{2} \)
3 \( 1 - 3.67e16T + 8.34e32T^{2} \)
5 \( 1 + 2.01e24T + 1.69e48T^{2} \)
7 \( 1 - 2.05e29T + 2.05e58T^{2} \)
11 \( 1 + 1.09e36T + 7.17e71T^{2} \)
13 \( 1 + 2.46e37T + 7.27e76T^{2} \)
17 \( 1 + 2.55e42T + 7.96e84T^{2} \)
19 \( 1 + 1.09e44T + 1.71e88T^{2} \)
23 \( 1 + 8.34e45T + 9.10e93T^{2} \)
29 \( 1 + 4.43e49T + 8.04e100T^{2} \)
31 \( 1 + 2.07e51T + 8.01e102T^{2} \)
37 \( 1 - 2.01e54T + 1.60e108T^{2} \)
41 \( 1 + 1.70e55T + 1.91e111T^{2} \)
43 \( 1 + 1.46e55T + 5.12e112T^{2} \)
47 \( 1 + 7.82e57T + 2.37e115T^{2} \)
53 \( 1 - 4.83e59T + 9.44e118T^{2} \)
59 \( 1 + 1.34e61T + 1.54e122T^{2} \)
61 \( 1 - 3.06e61T + 1.54e123T^{2} \)
67 \( 1 - 5.79e62T + 9.98e125T^{2} \)
71 \( 1 - 5.96e63T + 5.45e127T^{2} \)
73 \( 1 + 2.58e63T + 3.70e128T^{2} \)
79 \( 1 + 1.36e65T + 8.63e130T^{2} \)
83 \( 1 - 4.54e65T + 2.60e132T^{2} \)
89 \( 1 + 6.95e66T + 3.22e134T^{2} \)
97 \( 1 - 3.36e68T + 1.22e137T^{2} \)
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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

−15.33142775789665464152808433329, −14.62144898066989195572976184089, −13.08286449419592919848122442194, −11.34282676636948805795166073956, −8.570633730192148129111991215485, −7.81097344317321505890215721680, −4.84086002681136528295761321043, −3.84643910427900934906866316490, −2.44941283318093265885292563093, 0, 2.44941283318093265885292563093, 3.84643910427900934906866316490, 4.84086002681136528295761321043, 7.81097344317321505890215721680, 8.570633730192148129111991215485, 11.34282676636948805795166073956, 13.08286449419592919848122442194, 14.62144898066989195572976184089, 15.33142775789665464152808433329

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