Properties

Label 2-1-1.1-c39-0-1
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  + 2.41e5·2-s − 3.14e9·3-s − 4.91e11·4-s + 5.36e13·5-s − 7.60e14·6-s + 1.82e16·7-s − 2.51e17·8-s + 5.86e18·9-s + 1.29e19·10-s + 2.56e20·11-s + 1.54e21·12-s − 2.93e21·13-s + 4.39e21·14-s − 1.68e23·15-s + 2.09e23·16-s − 9.89e22·17-s + 1.41e24·18-s + 1.24e25·19-s − 2.63e25·20-s − 5.73e25·21-s + 6.20e25·22-s − 1.86e26·23-s + 7.91e26·24-s + 1.05e27·25-s − 7.08e26·26-s − 5.71e27·27-s − 8.94e27·28-s + ⋯
L(s)  = 1  + 0.325·2-s − 1.56·3-s − 0.893·4-s + 1.25·5-s − 0.509·6-s + 0.603·7-s − 0.616·8-s + 1.44·9-s + 0.409·10-s + 1.26·11-s + 1.39·12-s − 0.556·13-s + 0.196·14-s − 1.96·15-s + 0.693·16-s − 0.100·17-s + 0.471·18-s + 1.44·19-s − 1.12·20-s − 0.944·21-s + 0.412·22-s − 0.520·23-s + 0.965·24-s + 0.580·25-s − 0.181·26-s − 0.700·27-s − 0.539·28-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

\(L(20)\) \(\approx\) \(1.307032683\)
\(L(\frac12)\) \(\approx\) \(1.307032683\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.41e5T + 5.49e11T^{2} \)
3 \( 1 + 3.14e9T + 4.05e18T^{2} \)
5 \( 1 - 5.36e13T + 1.81e27T^{2} \)
7 \( 1 - 1.82e16T + 9.09e32T^{2} \)
11 \( 1 - 2.56e20T + 4.11e40T^{2} \)
13 \( 1 + 2.93e21T + 2.77e43T^{2} \)
17 \( 1 + 9.89e22T + 9.71e47T^{2} \)
19 \( 1 - 1.24e25T + 7.43e49T^{2} \)
23 \( 1 + 1.86e26T + 1.28e53T^{2} \)
29 \( 1 - 1.31e28T + 1.08e57T^{2} \)
31 \( 1 - 1.89e29T + 1.45e58T^{2} \)
37 \( 1 - 3.22e30T + 1.44e61T^{2} \)
41 \( 1 + 2.27e31T + 7.91e62T^{2} \)
43 \( 1 - 5.36e31T + 5.07e63T^{2} \)
47 \( 1 - 1.87e32T + 1.62e65T^{2} \)
53 \( 1 + 3.22e32T + 1.76e67T^{2} \)
59 \( 1 - 1.01e34T + 1.15e69T^{2} \)
61 \( 1 + 8.66e34T + 4.24e69T^{2} \)
67 \( 1 + 5.98e33T + 1.64e71T^{2} \)
71 \( 1 - 5.02e35T + 1.58e72T^{2} \)
73 \( 1 - 1.14e36T + 4.67e72T^{2} \)
79 \( 1 - 7.78e36T + 1.01e74T^{2} \)
83 \( 1 + 4.09e37T + 6.98e74T^{2} \)
89 \( 1 - 2.04e38T + 1.06e76T^{2} \)
97 \( 1 + 2.29e38T + 3.04e77T^{2} \)
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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

−22.59011503865665039268116986370, −21.70008808093715722491438337405, −17.95800440958010677951734099586, −17.15956122238140913271036887113, −13.97600905151550172193536490921, −11.95568779109767373812831894583, −9.745579757749797765508232032316, −6.06969296648389121288607488698, −4.80382247805960569135205656045, −1.05083340459810309799694031290, 1.05083340459810309799694031290, 4.80382247805960569135205656045, 6.06969296648389121288607488698, 9.745579757749797765508232032316, 11.95568779109767373812831894583, 13.97600905151550172193536490921, 17.15956122238140913271036887113, 17.95800440958010677951734099586, 21.70008808093715722491438337405, 22.59011503865665039268116986370

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