Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 67
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.06e9·2-s − 7.82e15·3-s − 1.21e20·4-s − 8.33e22·5-s − 3.96e25·6-s − 1.15e28·7-s − 1.36e30·8-s − 3.14e31·9-s − 4.22e32·10-s − 1.09e35·11-s + 9.53e35·12-s + 1.54e37·13-s − 5.83e37·14-s + 6.52e38·15-s + 1.10e40·16-s + 8.24e40·17-s − 1.59e41·18-s − 5.93e42·19-s + 1.01e43·20-s + 9.00e43·21-s − 5.53e44·22-s + 2.28e45·23-s + 1.06e46·24-s − 6.08e46·25-s + 7.81e46·26-s + 9.71e47·27-s + 1.40e48·28-s + ⋯
L(s)  = 1  + 0.417·2-s − 0.812·3-s − 0.826·4-s − 0.320·5-s − 0.338·6-s − 0.562·7-s − 0.761·8-s − 0.339·9-s − 0.133·10-s − 1.41·11-s + 0.671·12-s + 0.743·13-s − 0.234·14-s + 0.260·15-s + 0.508·16-s + 0.496·17-s − 0.141·18-s − 0.861·19-s + 0.264·20-s + 0.457·21-s − 0.591·22-s + 0.550·23-s + 0.618·24-s − 0.897·25-s + 0.310·26-s + 1.08·27-s + 0.464·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(67\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1,\ (\ :67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(0.5873099641\)
\(L(\frac12)\)  \(\approx\)  \(0.5873099641\)
\(L(\frac{69}{2})\)   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 \( 1 - 5.06e9T + 1.47e20T^{2} \)
3 \( 1 + 7.82e15T + 9.27e31T^{2} \)
5 \( 1 + 8.33e22T + 6.77e46T^{2} \)
7 \( 1 + 1.15e28T + 4.18e56T^{2} \)
11 \( 1 + 1.09e35T + 5.93e69T^{2} \)
13 \( 1 - 1.54e37T + 4.30e74T^{2} \)
17 \( 1 - 8.24e40T + 2.75e82T^{2} \)
19 \( 1 + 5.93e42T + 4.74e85T^{2} \)
23 \( 1 - 2.28e45T + 1.72e91T^{2} \)
29 \( 1 - 1.07e49T + 9.56e97T^{2} \)
31 \( 1 - 1.32e50T + 8.34e99T^{2} \)
37 \( 1 + 5.20e52T + 1.17e105T^{2} \)
41 \( 1 + 1.63e54T + 1.13e108T^{2} \)
43 \( 1 - 5.64e54T + 2.76e109T^{2} \)
47 \( 1 - 2.41e55T + 1.07e112T^{2} \)
53 \( 1 - 2.84e57T + 3.36e115T^{2} \)
59 \( 1 + 3.30e59T + 4.43e118T^{2} \)
61 \( 1 + 6.89e59T + 4.14e119T^{2} \)
67 \( 1 - 2.58e61T + 2.22e122T^{2} \)
71 \( 1 + 1.34e62T + 1.08e124T^{2} \)
73 \( 1 + 1.11e62T + 6.96e124T^{2} \)
79 \( 1 - 1.23e63T + 1.38e127T^{2} \)
83 \( 1 - 3.41e63T + 3.78e128T^{2} \)
89 \( 1 - 3.04e65T + 4.06e130T^{2} \)
97 \( 1 - 2.29e66T + 1.29e133T^{2} \)
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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

−17.43012023292266600367403828535, −15.65853967129481623159398249662, −13.60982873229299309727506926887, −12.22874343166330535552594730334, −10.41186364793203780853614668045, −8.398297559593765716015857212247, −6.09078674297176093913354331349, −4.85449009180606295178890287890, −3.15911341946688949251242986902, −0.46173712903443472817543896296, 0.46173712903443472817543896296, 3.15911341946688949251242986902, 4.85449009180606295178890287890, 6.09078674297176093913354331349, 8.398297559593765716015857212247, 10.41186364793203780853614668045, 12.22874343166330535552594730334, 13.60982873229299309727506926887, 15.65853967129481623159398249662, 17.43012023292266600367403828535

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