Properties

Label 2-23-1.1-c11-0-5
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $17.6718$
Root an. cond. $4.20379$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 69.2·2-s − 806.·3-s + 2.74e3·4-s − 1.05e4·5-s + 5.58e4·6-s − 3.92e4·7-s − 4.82e4·8-s + 4.72e5·9-s + 7.31e5·10-s + 5.62e5·11-s − 2.21e6·12-s − 7.14e5·13-s + 2.71e6·14-s + 8.51e6·15-s − 2.27e6·16-s − 4.57e6·17-s − 3.27e7·18-s + 4.10e6·19-s − 2.90e7·20-s + 3.16e7·21-s − 3.89e7·22-s + 6.43e6·23-s + 3.89e7·24-s + 6.28e7·25-s + 4.94e7·26-s − 2.38e8·27-s − 1.07e8·28-s + ⋯
L(s)  = 1  − 1.52·2-s − 1.91·3-s + 1.34·4-s − 1.51·5-s + 2.93·6-s − 0.882·7-s − 0.521·8-s + 2.66·9-s + 2.31·10-s + 1.05·11-s − 2.56·12-s − 0.533·13-s + 1.35·14-s + 2.89·15-s − 0.543·16-s − 0.782·17-s − 4.08·18-s + 0.380·19-s − 2.02·20-s + 1.69·21-s − 1.61·22-s + 0.208·23-s + 0.998·24-s + 1.28·25-s + 0.816·26-s − 3.19·27-s − 1.18·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(17.6718\)
Root analytic conductor: \(4.20379\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - 6.43e6T \)
good2 \( 1 + 69.2T + 2.04e3T^{2} \)
3 \( 1 + 806.T + 1.77e5T^{2} \)
5 \( 1 + 1.05e4T + 4.88e7T^{2} \)
7 \( 1 + 3.92e4T + 1.97e9T^{2} \)
11 \( 1 - 5.62e5T + 2.85e11T^{2} \)
13 \( 1 + 7.14e5T + 1.79e12T^{2} \)
17 \( 1 + 4.57e6T + 3.42e13T^{2} \)
19 \( 1 - 4.10e6T + 1.16e14T^{2} \)
29 \( 1 - 1.74e8T + 1.22e16T^{2} \)
31 \( 1 + 2.09e8T + 2.54e16T^{2} \)
37 \( 1 - 3.14e8T + 1.77e17T^{2} \)
41 \( 1 - 5.92e8T + 5.50e17T^{2} \)
43 \( 1 - 4.99e8T + 9.29e17T^{2} \)
47 \( 1 - 2.05e9T + 2.47e18T^{2} \)
53 \( 1 + 1.05e9T + 9.26e18T^{2} \)
59 \( 1 - 9.13e7T + 3.01e19T^{2} \)
61 \( 1 + 9.65e9T + 4.35e19T^{2} \)
67 \( 1 - 1.15e10T + 1.22e20T^{2} \)
71 \( 1 + 5.03e9T + 2.31e20T^{2} \)
73 \( 1 + 1.30e10T + 3.13e20T^{2} \)
79 \( 1 - 3.77e10T + 7.47e20T^{2} \)
83 \( 1 - 8.51e9T + 1.28e21T^{2} \)
89 \( 1 + 5.24e10T + 2.77e21T^{2} \)
97 \( 1 + 7.95e10T + 7.15e21T^{2} \)
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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

−15.62289246503140552585928969759, −12.46542599954661207217839977698, −11.57655336308808370151234035005, −10.70753770721537750544560609360, −9.350897528248305854084951886462, −7.42917209220168957904522653755, −6.52967549761884850141539522426, −4.33139547589029234116406760361, −0.880698169612953561047549875155, 0, 0.880698169612953561047549875155, 4.33139547589029234116406760361, 6.52967549761884850141539522426, 7.42917209220168957904522653755, 9.350897528248305854084951886462, 10.70753770721537750544560609360, 11.57655336308808370151234035005, 12.46542599954661207217839977698, 15.62289246503140552585928969759

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