Properties

Label 2-23-1.1-c11-0-15
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  + 28.6·2-s + 141.·3-s − 1.22e3·4-s − 2.64e3·5-s + 4.04e3·6-s + 7.64e4·7-s − 9.37e4·8-s − 1.57e5·9-s − 7.57e4·10-s − 6.03e5·11-s − 1.73e5·12-s − 9.41e5·13-s + 2.18e6·14-s − 3.74e5·15-s − 1.65e5·16-s − 5.54e6·17-s − 4.49e6·18-s − 8.15e6·19-s + 3.25e6·20-s + 1.08e7·21-s − 1.72e7·22-s + 6.43e6·23-s − 1.32e7·24-s − 4.18e7·25-s − 2.69e7·26-s − 4.72e7·27-s − 9.39e7·28-s + ⋯
L(s)  = 1  + 0.632·2-s + 0.336·3-s − 0.600·4-s − 0.378·5-s + 0.212·6-s + 1.71·7-s − 1.01·8-s − 0.887·9-s − 0.239·10-s − 1.12·11-s − 0.201·12-s − 0.703·13-s + 1.08·14-s − 0.127·15-s − 0.0394·16-s − 0.947·17-s − 0.560·18-s − 0.755·19-s + 0.227·20-s + 0.577·21-s − 0.714·22-s + 0.208·23-s − 0.340·24-s − 0.856·25-s − 0.444·26-s − 0.634·27-s − 1.03·28-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 \( 1 - 28.6T + 2.04e3T^{2} \)
3 \( 1 - 141.T + 1.77e5T^{2} \)
5 \( 1 + 2.64e3T + 4.88e7T^{2} \)
7 \( 1 - 7.64e4T + 1.97e9T^{2} \)
11 \( 1 + 6.03e5T + 2.85e11T^{2} \)
13 \( 1 + 9.41e5T + 1.79e12T^{2} \)
17 \( 1 + 5.54e6T + 3.42e13T^{2} \)
19 \( 1 + 8.15e6T + 1.16e14T^{2} \)
29 \( 1 - 1.14e8T + 1.22e16T^{2} \)
31 \( 1 + 1.09e8T + 2.54e16T^{2} \)
37 \( 1 + 3.97e8T + 1.77e17T^{2} \)
41 \( 1 - 7.31e8T + 5.50e17T^{2} \)
43 \( 1 - 7.98e8T + 9.29e17T^{2} \)
47 \( 1 + 4.30e8T + 2.47e18T^{2} \)
53 \( 1 - 4.54e9T + 9.26e18T^{2} \)
59 \( 1 + 7.83e8T + 3.01e19T^{2} \)
61 \( 1 + 7.23e9T + 4.35e19T^{2} \)
67 \( 1 + 1.06e10T + 1.22e20T^{2} \)
71 \( 1 + 1.08e10T + 2.31e20T^{2} \)
73 \( 1 - 3.25e10T + 3.13e20T^{2} \)
79 \( 1 + 5.15e10T + 7.47e20T^{2} \)
83 \( 1 - 6.27e10T + 1.28e21T^{2} \)
89 \( 1 + 2.11e10T + 2.77e21T^{2} \)
97 \( 1 + 1.36e11T + 7.15e21T^{2} \)
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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

−14.52076073145702315096612398040, −13.57002958266378082813427963259, −12.10456306479385019411504751416, −10.85900652027330498347477049662, −8.799997257620092501523482552146, −7.87321887208697347504487800387, −5.41065272263762032386932266019, −4.36366449123772299926770985435, −2.44957969283203089533100686772, 0, 2.44957969283203089533100686772, 4.36366449123772299926770985435, 5.41065272263762032386932266019, 7.87321887208697347504487800387, 8.799997257620092501523482552146, 10.85900652027330498347477049662, 12.10456306479385019411504751416, 13.57002958266378082813427963259, 14.52076073145702315096612398040

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