Properties

Label 2-1-1.1-c39-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $9.63395$
Root an. cond. $3.10386$
Motivic weight $39$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.99e5·2-s + 1.27e9·3-s + 8.88e10·4-s − 6.16e13·5-s − 1.01e15·6-s + 1.43e16·7-s + 3.68e17·8-s − 2.43e18·9-s + 4.92e19·10-s + 3.35e20·11-s + 1.12e20·12-s + 3.01e21·13-s − 1.14e22·14-s − 7.83e22·15-s − 3.43e23·16-s + 1.44e24·17-s + 1.94e24·18-s − 1.59e24·19-s − 5.47e24·20-s + 1.82e25·21-s − 2.68e26·22-s + 2.74e26·23-s + 4.68e26·24-s + 1.98e27·25-s − 2.40e27·26-s − 8.24e27·27-s + 1.27e27·28-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.631·3-s + 0.161·4-s − 1.44·5-s − 0.680·6-s + 0.476·7-s + 0.903·8-s − 0.601·9-s + 1.55·10-s + 1.65·11-s + 0.102·12-s + 0.571·13-s − 0.513·14-s − 0.912·15-s − 1.13·16-s + 1.46·17-s + 0.648·18-s − 0.185·19-s − 0.233·20-s + 0.301·21-s − 1.78·22-s + 0.767·23-s + 0.570·24-s + 1.08·25-s − 0.616·26-s − 1.01·27-s + 0.0770·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(40-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+39/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(9.63395\)
Root analytic conductor: \(3.10386\)
Motivic weight: \(39\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(0.9480966693\)
\(L(\frac12)\) \(\approx\) \(0.9480966693\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 7.99e5T + 5.49e11T^{2} \)
3 \( 1 - 1.27e9T + 4.05e18T^{2} \)
5 \( 1 + 6.16e13T + 1.81e27T^{2} \)
7 \( 1 - 1.43e16T + 9.09e32T^{2} \)
11 \( 1 - 3.35e20T + 4.11e40T^{2} \)
13 \( 1 - 3.01e21T + 2.77e43T^{2} \)
17 \( 1 - 1.44e24T + 9.71e47T^{2} \)
19 \( 1 + 1.59e24T + 7.43e49T^{2} \)
23 \( 1 - 2.74e26T + 1.28e53T^{2} \)
29 \( 1 + 8.78e27T + 1.08e57T^{2} \)
31 \( 1 + 4.55e28T + 1.45e58T^{2} \)
37 \( 1 - 5.60e30T + 1.44e61T^{2} \)
41 \( 1 - 9.03e30T + 7.91e62T^{2} \)
43 \( 1 + 1.90e31T + 5.07e63T^{2} \)
47 \( 1 - 6.24e32T + 1.62e65T^{2} \)
53 \( 1 - 4.67e33T + 1.76e67T^{2} \)
59 \( 1 + 1.88e34T + 1.15e69T^{2} \)
61 \( 1 - 5.59e34T + 4.24e69T^{2} \)
67 \( 1 + 4.11e34T + 1.64e71T^{2} \)
71 \( 1 - 9.44e35T + 1.58e72T^{2} \)
73 \( 1 + 5.48e35T + 4.67e72T^{2} \)
79 \( 1 + 8.45e36T + 1.01e74T^{2} \)
83 \( 1 - 8.82e36T + 6.98e74T^{2} \)
89 \( 1 - 1.09e38T + 1.06e76T^{2} \)
97 \( 1 - 3.24e38T + 3.04e77T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.06356577365630512087122742503, −20.04297962826433360525740266240, −18.99168337441610681374398997013, −16.80614908201608566134917323488, −14.55954964682038671985392725187, −11.46311048102242835954463911094, −8.939383381523033064774081454680, −7.74218623091617256182582349878, −3.83192808672040979997988668696, −0.966466068075954900303515791873, 0.966466068075954900303515791873, 3.83192808672040979997988668696, 7.74218623091617256182582349878, 8.939383381523033064774081454680, 11.46311048102242835954463911094, 14.55954964682038671985392725187, 16.80614908201608566134917323488, 18.99168337441610681374398997013, 20.04297962826433360525740266240, 23.06356577365630512087122742503

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