Properties

Label 2-1-1.1-c23-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $3.35204$
Root an. cond. $1.83085$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.01e3·2-s + 3.88e5·3-s + 7.74e6·4-s + 1.05e8·5-s − 1.56e9·6-s + 3.81e9·7-s + 2.59e9·8-s + 5.67e10·9-s − 4.21e11·10-s + 2.52e11·11-s + 3.00e12·12-s − 3.59e12·13-s − 1.53e13·14-s + 4.08e13·15-s − 7.53e13·16-s + 2.34e14·17-s − 2.27e14·18-s − 6.23e14·19-s + 8.13e14·20-s + 1.48e15·21-s − 1.01e15·22-s − 3.58e15·23-s + 1.00e15·24-s − 8.82e14·25-s + 1.44e16·26-s − 1.45e16·27-s + 2.95e16·28-s + ⋯
L(s)  = 1  − 1.38·2-s + 1.26·3-s + 0.922·4-s + 0.962·5-s − 1.75·6-s + 0.728·7-s + 0.106·8-s + 0.602·9-s − 1.33·10-s + 0.266·11-s + 1.16·12-s − 0.555·13-s − 1.01·14-s + 1.21·15-s − 1.07·16-s + 1.65·17-s − 0.835·18-s − 1.22·19-s + 0.888·20-s + 0.922·21-s − 0.369·22-s − 0.785·23-s + 0.135·24-s − 0.0740·25-s + 0.770·26-s − 0.502·27-s + 0.672·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(24-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+23/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(3.35204\)
Root analytic conductor: \(1.83085\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(1.219462729\)
\(L(\frac12)\) \(\approx\) \(1.219462729\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 4.01e3T + 8.38e6T^{2} \)
3 \( 1 - 3.88e5T + 9.41e10T^{2} \)
5 \( 1 - 1.05e8T + 1.19e16T^{2} \)
7 \( 1 - 3.81e9T + 2.73e19T^{2} \)
11 \( 1 - 2.52e11T + 8.95e23T^{2} \)
13 \( 1 + 3.59e12T + 4.17e25T^{2} \)
17 \( 1 - 2.34e14T + 1.99e28T^{2} \)
19 \( 1 + 6.23e14T + 2.57e29T^{2} \)
23 \( 1 + 3.58e15T + 2.08e31T^{2} \)
29 \( 1 + 2.05e16T + 4.31e33T^{2} \)
31 \( 1 - 1.36e17T + 2.00e34T^{2} \)
37 \( 1 + 1.23e18T + 1.17e36T^{2} \)
41 \( 1 - 1.40e18T + 1.24e37T^{2} \)
43 \( 1 - 2.18e17T + 3.71e37T^{2} \)
47 \( 1 + 8.67e18T + 2.87e38T^{2} \)
53 \( 1 + 7.63e19T + 4.55e39T^{2} \)
59 \( 1 + 1.01e18T + 5.36e40T^{2} \)
61 \( 1 - 2.87e20T + 1.15e41T^{2} \)
67 \( 1 - 1.47e21T + 9.99e41T^{2} \)
71 \( 1 - 7.64e20T + 3.79e42T^{2} \)
73 \( 1 + 3.49e21T + 7.18e42T^{2} \)
79 \( 1 - 1.02e22T + 4.42e43T^{2} \)
83 \( 1 + 7.71e21T + 1.37e44T^{2} \)
89 \( 1 - 4.58e21T + 6.85e44T^{2} \)
97 \( 1 + 1.13e23T + 4.96e45T^{2} \)
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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

−27.71975392964510574357267300963, −26.08002412254356619214664405113, −25.06262244345535832584628889273, −20.93411007176618065710844759575, −19.18449488041112370031013694544, −17.33046253304659494499678304849, −14.24685729890788642043577439014, −9.805490244224172864903671042701, −8.150088233572981620130000328332, −1.88602626271288641226254344200, 1.88602626271288641226254344200, 8.150088233572981620130000328332, 9.805490244224172864903671042701, 14.24685729890788642043577439014, 17.33046253304659494499678304849, 19.18449488041112370031013694544, 20.93411007176618065710844759575, 25.06262244345535832584628889273, 26.08002412254356619214664405113, 27.71975392964510574357267300963

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