Properties

Label 2-23-1.1-c13-0-18
Degree $2$
Conductor $23$
Sign $1$
Analytic cond. $24.6631$
Root an. cond. $4.96619$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 145.·2-s + 943.·3-s + 1.29e4·4-s + 5.80e4·5-s + 1.37e5·6-s + 3.92e5·7-s + 6.91e5·8-s − 7.04e5·9-s + 8.44e6·10-s − 8.47e6·11-s + 1.22e7·12-s − 2.22e7·13-s + 5.71e7·14-s + 5.47e7·15-s − 5.55e6·16-s − 1.37e7·17-s − 1.02e8·18-s + 2.81e8·19-s + 7.51e8·20-s + 3.70e8·21-s − 1.23e9·22-s + 1.48e8·23-s + 6.51e8·24-s + 2.15e9·25-s − 3.22e9·26-s − 2.16e9·27-s + 5.08e9·28-s + ⋯
L(s)  = 1  + 1.60·2-s + 0.746·3-s + 1.58·4-s + 1.66·5-s + 1.19·6-s + 1.26·7-s + 0.932·8-s − 0.442·9-s + 2.67·10-s − 1.44·11-s + 1.18·12-s − 1.27·13-s + 2.02·14-s + 1.24·15-s − 0.0828·16-s − 0.138·17-s − 0.710·18-s + 1.37·19-s + 2.62·20-s + 0.942·21-s − 2.31·22-s + 0.208·23-s + 0.696·24-s + 1.76·25-s − 2.04·26-s − 1.07·27-s + 1.99·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $1$
Analytic conductor: \(24.6631\)
Root analytic conductor: \(4.96619\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(7.516186501\)
\(L(\frac12)\) \(\approx\) \(7.516186501\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - 1.48e8T \)
good2 \( 1 - 145.T + 8.19e3T^{2} \)
3 \( 1 - 943.T + 1.59e6T^{2} \)
5 \( 1 - 5.80e4T + 1.22e9T^{2} \)
7 \( 1 - 3.92e5T + 9.68e10T^{2} \)
11 \( 1 + 8.47e6T + 3.45e13T^{2} \)
13 \( 1 + 2.22e7T + 3.02e14T^{2} \)
17 \( 1 + 1.37e7T + 9.90e15T^{2} \)
19 \( 1 - 2.81e8T + 4.20e16T^{2} \)
29 \( 1 + 4.81e9T + 1.02e19T^{2} \)
31 \( 1 - 3.09e9T + 2.44e19T^{2} \)
37 \( 1 - 5.68e9T + 2.43e20T^{2} \)
41 \( 1 + 5.28e9T + 9.25e20T^{2} \)
43 \( 1 - 6.74e10T + 1.71e21T^{2} \)
47 \( 1 - 2.35e10T + 5.46e21T^{2} \)
53 \( 1 + 1.46e11T + 2.60e22T^{2} \)
59 \( 1 + 5.20e11T + 1.04e23T^{2} \)
61 \( 1 - 1.88e11T + 1.61e23T^{2} \)
67 \( 1 - 4.58e11T + 5.48e23T^{2} \)
71 \( 1 - 1.49e12T + 1.16e24T^{2} \)
73 \( 1 - 1.78e12T + 1.67e24T^{2} \)
79 \( 1 + 8.60e10T + 4.66e24T^{2} \)
83 \( 1 + 3.98e12T + 8.87e24T^{2} \)
89 \( 1 + 1.67e12T + 2.19e25T^{2} \)
97 \( 1 + 1.35e12T + 6.73e25T^{2} \)
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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

−14.25644022342651308803269738576, −13.84786542099779655752614781426, −12.73763984965428977694552672003, −11.13021748038112644634837002328, −9.501184926969849781992656668839, −7.63986313515055384112587171225, −5.61604814862824404836861897773, −4.99384643220210869209398528195, −2.78824484374630204477591326622, −2.06732128767604962689498984082, 2.06732128767604962689498984082, 2.78824484374630204477591326622, 4.99384643220210869209398528195, 5.61604814862824404836861897773, 7.63986313515055384112587171225, 9.501184926969849781992656668839, 11.13021748038112644634837002328, 12.73763984965428977694552672003, 13.84786542099779655752614781426, 14.25644022342651308803269738576

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