Properties

Label 2-1-1.1-c45-0-2
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $12.8255$
Root an. cond. $3.58128$
Motivic weight $45$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.86e6·2-s + 8.00e10·3-s − 2.69e13·4-s − 9.35e15·5-s + 2.29e17·6-s − 6.95e18·7-s − 1.77e20·8-s + 3.46e21·9-s − 2.67e22·10-s + 9.22e22·11-s − 2.16e24·12-s − 7.74e24·13-s − 1.98e25·14-s − 7.49e26·15-s + 4.40e26·16-s − 1.39e27·17-s + 9.90e27·18-s + 3.42e28·19-s + 2.52e29·20-s − 5.56e29·21-s + 2.63e29·22-s − 4.51e30·23-s − 1.42e31·24-s + 5.90e31·25-s − 2.21e31·26-s + 4.06e31·27-s + 1.87e32·28-s + ⋯
L(s)  = 1  + 0.482·2-s + 1.47·3-s − 0.767·4-s − 1.75·5-s + 0.710·6-s − 0.671·7-s − 0.852·8-s + 1.17·9-s − 0.846·10-s + 0.341·11-s − 1.13·12-s − 0.669·13-s − 0.324·14-s − 2.58·15-s + 0.356·16-s − 0.287·17-s + 0.565·18-s + 0.579·19-s + 1.34·20-s − 0.990·21-s + 0.164·22-s − 1.03·23-s − 1.25·24-s + 2.07·25-s − 0.322·26-s + 0.252·27-s + 0.515·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(46-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+45/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(12.8255\)
Root analytic conductor: \(3.58128\)
Motivic weight: \(45\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :45/2),\ -1)\)

Particular Values

\(L(23)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{47}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.86e6T + 3.51e13T^{2} \)
3 \( 1 - 8.00e10T + 2.95e21T^{2} \)
5 \( 1 + 9.35e15T + 2.84e31T^{2} \)
7 \( 1 + 6.95e18T + 1.07e38T^{2} \)
11 \( 1 - 9.22e22T + 7.28e46T^{2} \)
13 \( 1 + 7.74e24T + 1.34e50T^{2} \)
17 \( 1 + 1.39e27T + 2.34e55T^{2} \)
19 \( 1 - 3.42e28T + 3.49e57T^{2} \)
23 \( 1 + 4.51e30T + 1.89e61T^{2} \)
29 \( 1 + 2.87e31T + 6.42e65T^{2} \)
31 \( 1 + 2.99e33T + 1.29e67T^{2} \)
37 \( 1 + 1.63e35T + 3.70e70T^{2} \)
41 \( 1 + 2.25e35T + 3.76e72T^{2} \)
43 \( 1 - 9.49e36T + 3.20e73T^{2} \)
47 \( 1 - 5.21e37T + 1.75e75T^{2} \)
53 \( 1 + 8.67e38T + 3.91e77T^{2} \)
59 \( 1 + 4.25e39T + 4.87e79T^{2} \)
61 \( 1 + 2.62e40T + 2.18e80T^{2} \)
67 \( 1 - 4.33e40T + 1.49e82T^{2} \)
71 \( 1 - 3.71e41T + 2.02e83T^{2} \)
73 \( 1 + 1.04e42T + 7.07e83T^{2} \)
79 \( 1 - 3.47e42T + 2.47e85T^{2} \)
83 \( 1 + 6.23e42T + 2.28e86T^{2} \)
89 \( 1 + 3.38e43T + 5.27e87T^{2} \)
97 \( 1 - 4.61e44T + 2.53e89T^{2} \)
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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

−20.05877251886094004363043174180, −19.02943616266343248258861587188, −15.56062319188556808650697312767, −14.21371923317822788965977870210, −12.40426487154665503106833194686, −9.146910085028181413545895132657, −7.71136879901769074032724678054, −4.15376264120983200749644933658, −3.18153651066159805009838408765, 0, 3.18153651066159805009838408765, 4.15376264120983200749644933658, 7.71136879901769074032724678054, 9.146910085028181413545895132657, 12.40426487154665503106833194686, 14.21371923317822788965977870210, 15.56062319188556808650697312767, 19.02943616266343248258861587188, 20.05877251886094004363043174180

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