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  − 1.83e10·2-s + 2.28e16·3-s − 2.52e20·4-s + 1.12e24·5-s − 4.19e26·6-s − 8.82e28·7-s + 1.54e31·8-s − 3.13e32·9-s − 2.06e34·10-s − 6.35e35·11-s − 5.76e36·12-s + 1.50e38·13-s + 1.62e39·14-s + 2.56e40·15-s − 1.35e41·16-s + 5.11e42·17-s + 5.75e42·18-s − 8.09e43·19-s − 2.83e44·20-s − 2.01e45·21-s + 1.16e46·22-s − 3.60e46·23-s + 3.53e47·24-s − 4.31e47·25-s − 2.75e48·26-s − 2.61e49·27-s + 2.22e49·28-s + ⋯
L(s)  = 1  − 0.756·2-s + 0.790·3-s − 0.427·4-s + 0.863·5-s − 0.597·6-s − 0.616·7-s + 1.08·8-s − 0.375·9-s − 0.652·10-s − 0.749·11-s − 0.338·12-s + 0.556·13-s + 0.466·14-s + 0.682·15-s − 0.389·16-s + 1.81·17-s + 0.283·18-s − 0.618·19-s − 0.369·20-s − 0.487·21-s + 0.566·22-s − 0.377·23-s + 0.853·24-s − 0.254·25-s − 0.420·26-s − 1.08·27-s + 0.263·28-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 \( 1 + 1.83e10T + 5.90e20T^{2} \)
3 \( 1 - 2.28e16T + 8.34e32T^{2} \)
5 \( 1 - 1.12e24T + 1.69e48T^{2} \)
7 \( 1 + 8.82e28T + 2.05e58T^{2} \)
11 \( 1 + 6.35e35T + 7.17e71T^{2} \)
13 \( 1 - 1.50e38T + 7.27e76T^{2} \)
17 \( 1 - 5.11e42T + 7.96e84T^{2} \)
19 \( 1 + 8.09e43T + 1.71e88T^{2} \)
23 \( 1 + 3.60e46T + 9.10e93T^{2} \)
29 \( 1 - 5.01e49T + 8.04e100T^{2} \)
31 \( 1 + 5.34e51T + 8.01e102T^{2} \)
37 \( 1 + 1.65e54T + 1.60e108T^{2} \)
41 \( 1 - 6.67e55T + 1.91e111T^{2} \)
43 \( 1 + 2.02e56T + 5.12e112T^{2} \)
47 \( 1 + 7.63e57T + 2.37e115T^{2} \)
53 \( 1 - 2.97e59T + 9.44e118T^{2} \)
59 \( 1 - 1.35e60T + 1.54e122T^{2} \)
61 \( 1 + 5.28e61T + 1.54e123T^{2} \)
67 \( 1 + 1.38e63T + 9.98e125T^{2} \)
71 \( 1 + 2.36e63T + 5.45e127T^{2} \)
73 \( 1 - 1.10e64T + 3.70e128T^{2} \)
79 \( 1 + 7.32e64T + 8.63e130T^{2} \)
83 \( 1 - 3.62e65T + 2.60e132T^{2} \)
89 \( 1 + 6.55e66T + 3.22e134T^{2} \)
97 \( 1 - 4.82e68T + 1.22e137T^{2} \)
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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

−16.58697338299518308172869164642, −14.29754782200406479452022291266, −13.11406041710355855835891327284, −10.26926668803176569134232743412, −9.167861774318045912540525546355, −7.87951929313009344124672408927, −5.62343814090929368146225190322, −3.38110934497104234346407917904, −1.74239420542143451773655268590, 0, 1.74239420542143451773655268590, 3.38110934497104234346407917904, 5.62343814090929368146225190322, 7.87951929313009344124672408927, 9.167861774318045912540525546355, 10.26926668803176569134232743412, 13.11406041710355855835891327284, 14.29754782200406479452022291266, 16.58697338299518308172869164642

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