Properties

Label 2-23-1.1-c7-0-10
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $7.18485$
Root an. cond. $2.68045$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.41·2-s − 20.5·3-s − 86.8·4-s + 258.·5-s − 131.·6-s − 1.37e3·7-s − 1.37e3·8-s − 1.76e3·9-s + 1.65e3·10-s − 628.·11-s + 1.78e3·12-s − 664.·13-s − 8.81e3·14-s − 5.30e3·15-s + 2.27e3·16-s − 1.65e4·17-s − 1.13e4·18-s + 4.01e4·19-s − 2.24e4·20-s + 2.82e4·21-s − 4.03e3·22-s + 1.21e4·23-s + 2.83e4·24-s − 1.13e4·25-s − 4.26e3·26-s + 8.11e4·27-s + 1.19e5·28-s + ⋯
L(s)  = 1  + 0.566·2-s − 0.439·3-s − 0.678·4-s + 0.924·5-s − 0.248·6-s − 1.51·7-s − 0.951·8-s − 0.807·9-s + 0.524·10-s − 0.142·11-s + 0.297·12-s − 0.0839·13-s − 0.859·14-s − 0.406·15-s + 0.139·16-s − 0.814·17-s − 0.457·18-s + 1.34·19-s − 0.627·20-s + 0.665·21-s − 0.0807·22-s + 0.208·23-s + 0.417·24-s − 0.144·25-s − 0.0475·26-s + 0.793·27-s + 1.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(7.18485\)
Root analytic conductor: \(2.68045\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - 1.21e4T \)
good2 \( 1 - 6.41T + 128T^{2} \)
3 \( 1 + 20.5T + 2.18e3T^{2} \)
5 \( 1 - 258.T + 7.81e4T^{2} \)
7 \( 1 + 1.37e3T + 8.23e5T^{2} \)
11 \( 1 + 628.T + 1.94e7T^{2} \)
13 \( 1 + 664.T + 6.27e7T^{2} \)
17 \( 1 + 1.65e4T + 4.10e8T^{2} \)
19 \( 1 - 4.01e4T + 8.93e8T^{2} \)
29 \( 1 + 1.08e5T + 1.72e10T^{2} \)
31 \( 1 - 2.28e5T + 2.75e10T^{2} \)
37 \( 1 + 5.75e5T + 9.49e10T^{2} \)
41 \( 1 + 4.25e5T + 1.94e11T^{2} \)
43 \( 1 + 4.33e5T + 2.71e11T^{2} \)
47 \( 1 - 5.47e5T + 5.06e11T^{2} \)
53 \( 1 + 1.27e6T + 1.17e12T^{2} \)
59 \( 1 + 8.69e5T + 2.48e12T^{2} \)
61 \( 1 - 1.45e6T + 3.14e12T^{2} \)
67 \( 1 + 2.47e6T + 6.06e12T^{2} \)
71 \( 1 + 3.74e6T + 9.09e12T^{2} \)
73 \( 1 - 2.20e6T + 1.10e13T^{2} \)
79 \( 1 + 2.99e6T + 1.92e13T^{2} \)
83 \( 1 - 7.14e6T + 2.71e13T^{2} \)
89 \( 1 - 9.99e4T + 4.42e13T^{2} \)
97 \( 1 + 1.23e7T + 8.07e13T^{2} \)
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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

−15.60562320746668056375669469190, −13.92577651928988430088935144720, −13.27883355497102146663595692823, −11.99498469062388091400857550184, −10.04750736737213245528878393955, −9.019838499030636337384547371009, −6.39885422662374251451656667521, −5.31965024070023853876465709875, −3.15282368904463550948938733752, 0, 3.15282368904463550948938733752, 5.31965024070023853876465709875, 6.39885422662374251451656667521, 9.019838499030636337384547371009, 10.04750736737213245528878393955, 11.99498469062388091400857550184, 13.27883355497102146663595692823, 13.92577651928988430088935144720, 15.60562320746668056375669469190

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