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Properties

Label 2-23-23.22-c14-0-15
Degree $2$
Conductor $23$
Sign $0.198 - 0.980i$
Analytic cond. $28.5956$
Root an. cond. $5.34749$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 37.1·2-s + 4.21e3·3-s − 1.50e4·4-s + 1.26e5i·5-s + 1.56e5·6-s − 1.01e6i·7-s − 1.16e6·8-s + 1.29e7·9-s + 4.69e6i·10-s + 2.72e7i·11-s − 6.32e7·12-s + 8.12e6·13-s − 3.77e7i·14-s + 5.32e8i·15-s + 2.02e8·16-s + 3.49e8i·17-s + ⋯
L(s)  = 1  + 0.290·2-s + 1.92·3-s − 0.915·4-s + 1.61i·5-s + 0.559·6-s − 1.23i·7-s − 0.555·8-s + 2.71·9-s + 0.469i·10-s + 1.39i·11-s − 1.76·12-s + 0.129·13-s − 0.358i·14-s + 3.11i·15-s + 0.754·16-s + 0.851i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(28.5956\)
Root analytic conductor: \(5.34749\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :7),\ 0.198 - 0.980i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(3.514915481\)
\(L(\frac12)\) \(\approx\) \(3.514915481\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (6.74e8 - 3.33e9i)T \)
good2 \( 1 - 37.1T + 1.63e4T^{2} \)
3 \( 1 - 4.21e3T + 4.78e6T^{2} \)
5 \( 1 - 1.26e5iT - 6.10e9T^{2} \)
7 \( 1 + 1.01e6iT - 6.78e11T^{2} \)
11 \( 1 - 2.72e7iT - 3.79e14T^{2} \)
13 \( 1 - 8.12e6T + 3.93e15T^{2} \)
17 \( 1 - 3.49e8iT - 1.68e17T^{2} \)
19 \( 1 - 4.58e8iT - 7.99e17T^{2} \)
29 \( 1 + 1.51e10T + 2.97e20T^{2} \)
31 \( 1 + 3.61e9T + 7.56e20T^{2} \)
37 \( 1 - 1.19e9iT - 9.01e21T^{2} \)
41 \( 1 + 7.65e9T + 3.79e22T^{2} \)
43 \( 1 - 3.94e10iT - 7.38e22T^{2} \)
47 \( 1 - 4.24e11T + 2.56e23T^{2} \)
53 \( 1 + 1.33e12iT - 1.37e24T^{2} \)
59 \( 1 - 3.11e12T + 6.19e24T^{2} \)
61 \( 1 + 3.13e12iT - 9.87e24T^{2} \)
67 \( 1 + 4.52e11iT - 3.67e25T^{2} \)
71 \( 1 - 1.17e13T + 8.27e25T^{2} \)
73 \( 1 - 5.81e11T + 1.22e26T^{2} \)
79 \( 1 + 2.55e13iT - 3.68e26T^{2} \)
83 \( 1 + 5.45e12iT - 7.36e26T^{2} \)
89 \( 1 + 3.32e13iT - 1.95e27T^{2} \)
97 \( 1 + 4.07e13iT - 6.52e27T^{2} \)
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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

−14.59310584355075330538496486505, −13.87860456460607041862746090763, −12.92894612571454699453561034251, −10.33381576124817617579865030391, −9.617461971977652286339801295292, −7.901364745201864282252546405244, −7.04536807885897834239158293148, −4.06313028224488574691058511804, −3.44494745847262993616652483081, −1.88235482714540757332824156800, 0.864899248324378550264419045720, 2.61507190718930716349679068273, 4.02219650844349368856334328253, 5.34274942390263168653325792023, 8.234698804085692397873757824688, 8.837061162096497540842101130679, 9.364568905747661115343806457536, 12.35203952855859086473892498995, 13.25373582356087955005270540392, 14.01369603396869854402999727584

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