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  − 4.75e10·2-s − 9.70e15·3-s + 1.67e21·4-s − 1.40e24·5-s + 4.61e26·6-s − 1.18e27·7-s − 5.13e31·8-s − 7.40e32·9-s + 6.69e34·10-s + 2.72e34·11-s − 1.62e37·12-s + 4.14e38·13-s + 5.63e37·14-s + 1.36e40·15-s + 1.45e42·16-s − 7.81e41·17-s + 3.51e43·18-s + 6.36e43·19-s − 2.35e45·20-s + 1.14e43·21-s − 1.29e45·22-s + 8.76e46·23-s + 4.98e47·24-s + 2.86e47·25-s − 1.96e49·26-s + 1.52e49·27-s − 1.97e48·28-s + ⋯
L(s)  = 1  − 1.95·2-s − 0.335·3-s + 2.82·4-s − 1.08·5-s + 0.657·6-s − 0.00827·7-s − 3.57·8-s − 0.887·9-s + 2.11·10-s + 0.0321·11-s − 0.950·12-s + 1.53·13-s + 0.0161·14-s + 0.363·15-s + 4.17·16-s − 0.277·17-s + 1.73·18-s + 0.486·19-s − 3.05·20-s + 0.00277·21-s − 0.0629·22-s + 0.918·23-s + 1.20·24-s + 0.169·25-s − 3.00·26-s + 0.634·27-s − 0.0234·28-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 \( 1 + 4.75e10T + 5.90e20T^{2} \)
3 \( 1 + 9.70e15T + 8.34e32T^{2} \)
5 \( 1 + 1.40e24T + 1.69e48T^{2} \)
7 \( 1 + 1.18e27T + 2.05e58T^{2} \)
11 \( 1 - 2.72e34T + 7.17e71T^{2} \)
13 \( 1 - 4.14e38T + 7.27e76T^{2} \)
17 \( 1 + 7.81e41T + 7.96e84T^{2} \)
19 \( 1 - 6.36e43T + 1.71e88T^{2} \)
23 \( 1 - 8.76e46T + 9.10e93T^{2} \)
29 \( 1 - 5.98e49T + 8.04e100T^{2} \)
31 \( 1 - 1.30e51T + 8.01e102T^{2} \)
37 \( 1 - 1.23e54T + 1.60e108T^{2} \)
41 \( 1 - 1.03e55T + 1.91e111T^{2} \)
43 \( 1 + 2.30e56T + 5.12e112T^{2} \)
47 \( 1 - 5.31e57T + 2.37e115T^{2} \)
53 \( 1 + 2.46e59T + 9.44e118T^{2} \)
59 \( 1 + 7.33e60T + 1.54e122T^{2} \)
61 \( 1 + 4.55e61T + 1.54e123T^{2} \)
67 \( 1 + 9.77e62T + 9.98e125T^{2} \)
71 \( 1 + 1.10e64T + 5.45e127T^{2} \)
73 \( 1 - 3.21e63T + 3.70e128T^{2} \)
79 \( 1 - 3.60e65T + 8.63e130T^{2} \)
83 \( 1 - 1.23e66T + 2.60e132T^{2} \)
89 \( 1 + 1.50e67T + 3.22e134T^{2} \)
97 \( 1 + 4.11e68T + 1.22e137T^{2} \)
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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

−16.58382736082503921643057794474, −15.45193245293100586751209020195, −11.70096424134044930969095884903, −10.88575632043078929050876297018, −8.919332482578869035286588611111, −7.85327507052930294636656172219, −6.27825358075793592534840856417, −3.10273195639955045316355560619, −1.12778507481101958553075888661, 0, 1.12778507481101958553075888661, 3.10273195639955045316355560619, 6.27825358075793592534840856417, 7.85327507052930294636656172219, 8.919332482578869035286588611111, 10.88575632043078929050876297018, 11.70096424134044930969095884903, 15.45193245293100586751209020195, 16.58382736082503921643057794474

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