Properties

Label 2-1-1.1-c23-0-1
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  + 5.09e3·2-s − 4.89e4·3-s + 1.75e7·4-s − 3.19e7·5-s − 2.49e8·6-s − 5.17e9·7-s + 4.68e10·8-s − 9.17e10·9-s − 1.63e11·10-s + 6.04e11·11-s − 8.61e11·12-s + 7.96e12·13-s − 2.63e13·14-s + 1.56e12·15-s + 9.13e13·16-s + 1.98e13·17-s − 4.67e14·18-s + 6.27e14·19-s − 5.62e14·20-s + 2.53e14·21-s + 3.08e15·22-s − 4.55e15·23-s − 2.29e15·24-s − 1.08e16·25-s + 4.06e16·26-s + 9.10e15·27-s − 9.09e16·28-s + ⋯
L(s)  = 1  + 1.75·2-s − 0.159·3-s + 2.09·4-s − 0.293·5-s − 0.280·6-s − 0.988·7-s + 1.92·8-s − 0.974·9-s − 0.515·10-s + 0.638·11-s − 0.334·12-s + 1.23·13-s − 1.73·14-s + 0.0467·15-s + 1.29·16-s + 0.140·17-s − 1.71·18-s + 1.23·19-s − 0.614·20-s + 0.157·21-s + 1.12·22-s − 0.997·23-s − 0.307·24-s − 0.914·25-s + 2.16·26-s + 0.315·27-s − 2.07·28-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

\(L(12)\) \(\approx\) \(2.933205477\)
\(L(\frac12)\) \(\approx\) \(2.933205477\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 5.09e3T + 8.38e6T^{2} \)
3 \( 1 + 4.89e4T + 9.41e10T^{2} \)
5 \( 1 + 3.19e7T + 1.19e16T^{2} \)
7 \( 1 + 5.17e9T + 2.73e19T^{2} \)
11 \( 1 - 6.04e11T + 8.95e23T^{2} \)
13 \( 1 - 7.96e12T + 4.17e25T^{2} \)
17 \( 1 - 1.98e13T + 1.99e28T^{2} \)
19 \( 1 - 6.27e14T + 2.57e29T^{2} \)
23 \( 1 + 4.55e15T + 2.08e31T^{2} \)
29 \( 1 - 4.14e16T + 4.31e33T^{2} \)
31 \( 1 - 1.35e15T + 2.00e34T^{2} \)
37 \( 1 - 3.41e17T + 1.17e36T^{2} \)
41 \( 1 + 3.69e18T + 1.24e37T^{2} \)
43 \( 1 + 1.96e18T + 3.71e37T^{2} \)
47 \( 1 - 2.44e19T + 2.87e38T^{2} \)
53 \( 1 + 6.39e19T + 4.55e39T^{2} \)
59 \( 1 - 2.81e20T + 5.36e40T^{2} \)
61 \( 1 + 4.67e20T + 1.15e41T^{2} \)
67 \( 1 - 2.77e20T + 9.99e41T^{2} \)
71 \( 1 - 2.29e21T + 3.79e42T^{2} \)
73 \( 1 + 4.56e21T + 7.18e42T^{2} \)
79 \( 1 + 3.99e21T + 4.42e43T^{2} \)
83 \( 1 - 1.45e22T + 1.37e44T^{2} \)
89 \( 1 - 1.80e21T + 6.85e44T^{2} \)
97 \( 1 - 8.25e22T + 4.96e45T^{2} \)
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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

−28.80701753675713985205512131895, −25.43480965722488614012096939507, −23.39018336851803561722438513387, −22.26073495304594102943229224580, −20.11436418542491146408551822951, −15.96477547641143977921582445988, −13.81955445804823991803132300494, −11.79514317423093929496864845919, −6.07058342244853426264178449878, −3.46542238195904954925506904775, 3.46542238195904954925506904775, 6.07058342244853426264178449878, 11.79514317423093929496864845919, 13.81955445804823991803132300494, 15.96477547641143977921582445988, 20.11436418542491146408551822951, 22.26073495304594102943229224580, 23.39018336851803561722438513387, 25.43480965722488614012096939507, 28.80701753675713985205512131895

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