Properties

Label 2-1-1.1-c45-0-0
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  − 6.41e6·2-s − 1.72e10·3-s + 5.97e12·4-s + 2.46e15·5-s + 1.10e17·6-s + 1.29e19·7-s + 1.87e20·8-s − 2.65e21·9-s − 1.58e22·10-s + 7.78e22·11-s − 1.02e23·12-s − 1.84e25·13-s − 8.27e25·14-s − 4.24e25·15-s − 1.41e27·16-s − 2.82e27·17-s + 1.70e28·18-s + 8.52e28·19-s + 1.47e28·20-s − 2.22e29·21-s − 4.99e29·22-s − 2.09e30·23-s − 3.22e30·24-s − 2.23e31·25-s + 1.18e32·26-s + 9.65e31·27-s + 7.70e31·28-s + ⋯
L(s)  = 1  − 1.08·2-s − 0.316·3-s + 0.169·4-s + 0.462·5-s + 0.342·6-s + 1.24·7-s + 0.897·8-s − 0.899·9-s − 0.500·10-s + 0.288·11-s − 0.0537·12-s − 1.59·13-s − 1.34·14-s − 0.146·15-s − 1.14·16-s − 0.584·17-s + 0.973·18-s + 1.44·19-s + 0.0785·20-s − 0.394·21-s − 0.311·22-s − 0.481·23-s − 0.284·24-s − 0.785·25-s + 1.72·26-s + 0.601·27-s + 0.211·28-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 \( 1 + 6.41e6T + 3.51e13T^{2} \)
3 \( 1 + 1.72e10T + 2.95e21T^{2} \)
5 \( 1 - 2.46e15T + 2.84e31T^{2} \)
7 \( 1 - 1.29e19T + 1.07e38T^{2} \)
11 \( 1 - 7.78e22T + 7.28e46T^{2} \)
13 \( 1 + 1.84e25T + 1.34e50T^{2} \)
17 \( 1 + 2.82e27T + 2.34e55T^{2} \)
19 \( 1 - 8.52e28T + 3.49e57T^{2} \)
23 \( 1 + 2.09e30T + 1.89e61T^{2} \)
29 \( 1 + 1.33e33T + 6.42e65T^{2} \)
31 \( 1 + 9.42e32T + 1.29e67T^{2} \)
37 \( 1 + 8.78e34T + 3.70e70T^{2} \)
41 \( 1 - 2.03e36T + 3.76e72T^{2} \)
43 \( 1 + 6.02e36T + 3.20e73T^{2} \)
47 \( 1 + 4.75e37T + 1.75e75T^{2} \)
53 \( 1 + 7.36e38T + 3.91e77T^{2} \)
59 \( 1 + 9.51e39T + 4.87e79T^{2} \)
61 \( 1 + 1.95e39T + 2.18e80T^{2} \)
67 \( 1 - 2.10e41T + 1.49e82T^{2} \)
71 \( 1 + 5.43e41T + 2.02e83T^{2} \)
73 \( 1 - 8.99e41T + 7.07e83T^{2} \)
79 \( 1 + 7.61e41T + 2.47e85T^{2} \)
83 \( 1 - 6.62e42T + 2.28e86T^{2} \)
89 \( 1 + 6.63e43T + 5.27e87T^{2} \)
97 \( 1 - 1.95e44T + 2.53e89T^{2} \)
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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

−20.00141330000586300776155390382, −17.93672447684804605224613643041, −17.05939314902054771368099854329, −14.23348000659179508911246174538, −11.38744322909410240225914390179, −9.497692399288722992339187258041, −7.75364568294789525381115787121, −5.09196327013335612925130720207, −1.82170644350664873402382721900, 0, 1.82170644350664873402382721900, 5.09196327013335612925130720207, 7.75364568294789525381115787121, 9.497692399288722992339187258041, 11.38744322909410240225914390179, 14.23348000659179508911246174538, 17.05939314902054771368099854329, 17.93672447684804605224613643041, 20.00141330000586300776155390382

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