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  + 3.33e10·2-s − 1.33e16·3-s + 5.24e20·4-s + 1.09e24·5-s − 4.44e26·6-s − 1.48e29·7-s − 2.20e30·8-s − 6.57e32·9-s + 3.64e34·10-s + 3.06e35·11-s − 6.98e36·12-s + 1.14e38·13-s − 4.96e39·14-s − 1.45e40·15-s − 3.83e41·16-s − 4.51e42·17-s − 2.19e43·18-s + 1.29e44·19-s + 5.73e44·20-s + 1.97e45·21-s + 1.02e46·22-s − 1.62e47·23-s + 2.93e46·24-s − 4.98e47·25-s + 3.83e48·26-s + 1.98e49·27-s − 7.79e49·28-s + ⋯
L(s)  = 1  + 1.37·2-s − 0.460·3-s + 0.888·4-s + 0.839·5-s − 0.633·6-s − 1.03·7-s − 0.153·8-s − 0.787·9-s + 1.15·10-s + 0.362·11-s − 0.409·12-s + 0.425·13-s − 1.42·14-s − 0.387·15-s − 1.09·16-s − 1.59·17-s − 1.08·18-s + 0.986·19-s + 0.746·20-s + 0.478·21-s + 0.497·22-s − 1.70·23-s + 0.0707·24-s − 0.294·25-s + 0.584·26-s + 0.824·27-s − 0.922·28-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 \( 1 - 3.33e10T + 5.90e20T^{2} \)
3 \( 1 + 1.33e16T + 8.34e32T^{2} \)
5 \( 1 - 1.09e24T + 1.69e48T^{2} \)
7 \( 1 + 1.48e29T + 2.05e58T^{2} \)
11 \( 1 - 3.06e35T + 7.17e71T^{2} \)
13 \( 1 - 1.14e38T + 7.27e76T^{2} \)
17 \( 1 + 4.51e42T + 7.96e84T^{2} \)
19 \( 1 - 1.29e44T + 1.71e88T^{2} \)
23 \( 1 + 1.62e47T + 9.10e93T^{2} \)
29 \( 1 + 5.30e50T + 8.04e100T^{2} \)
31 \( 1 - 1.03e51T + 8.01e102T^{2} \)
37 \( 1 - 3.96e53T + 1.60e108T^{2} \)
41 \( 1 - 3.97e55T + 1.91e111T^{2} \)
43 \( 1 - 2.19e56T + 5.12e112T^{2} \)
47 \( 1 + 1.05e57T + 2.37e115T^{2} \)
53 \( 1 + 7.74e58T + 9.44e118T^{2} \)
59 \( 1 - 4.93e60T + 1.54e122T^{2} \)
61 \( 1 - 4.27e61T + 1.54e123T^{2} \)
67 \( 1 - 1.10e63T + 9.98e125T^{2} \)
71 \( 1 + 1.05e64T + 5.45e127T^{2} \)
73 \( 1 - 3.43e64T + 3.70e128T^{2} \)
79 \( 1 + 3.22e65T + 8.63e130T^{2} \)
83 \( 1 + 1.44e66T + 2.60e132T^{2} \)
89 \( 1 - 1.81e67T + 3.22e134T^{2} \)
97 \( 1 - 3.55e68T + 1.22e137T^{2} \)
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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

−15.96122632515041024550109088627, −14.08661564886554384675425759181, −13.03507879826441797887384240436, −11.47112721253375282028815365013, −9.367970717525828935737881679877, −6.36950965213024748672164192648, −5.64015087618336625205524727929, −3.86294287331884541733476343675, −2.37531976789441885212991925895, 0, 2.37531976789441885212991925895, 3.86294287331884541733476343675, 5.64015087618336625205524727929, 6.36950965213024748672164192648, 9.367970717525828935737881679877, 11.47112721253375282028815365013, 13.03507879826441797887384240436, 14.08661564886554384675425759181, 15.96122632515041024550109088627

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