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Properties

Label 2-23-23.22-c24-0-18
Degree $2$
Conductor $23$
Sign $-0.919 - 0.393i$
Analytic cond. $83.9424$
Root an. cond. $9.16201$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.84e3·2-s − 7.96e5·3-s + 4.47e7·4-s + 2.92e8i·5-s − 6.24e9·6-s + 2.55e10i·7-s + 2.19e11·8-s + 3.51e11·9-s + 2.29e12i·10-s + 3.79e12i·11-s − 3.56e13·12-s + 1.53e12·13-s + 2.00e14i·14-s − 2.32e14i·15-s + 9.68e14·16-s − 5.00e14i·17-s + ⋯
L(s)  = 1  + 1.91·2-s − 1.49·3-s + 2.66·4-s + 1.19i·5-s − 2.86·6-s + 1.84i·7-s + 3.18·8-s + 1.24·9-s + 2.29i·10-s + 1.21i·11-s − 3.99·12-s + 0.0658·13-s + 3.54i·14-s − 1.79i·15-s + 3.44·16-s − 0.859i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.919 - 0.393i$
Analytic conductor: \(83.9424\)
Root analytic conductor: \(9.16201\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :12),\ -0.919 - 0.393i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(4.023109525\)
\(L(\frac12)\) \(\approx\) \(4.023109525\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (2.01e16 + 8.61e15i)T \)
good2 \( 1 - 7.84e3T + 1.67e7T^{2} \)
3 \( 1 + 7.96e5T + 2.82e11T^{2} \)
5 \( 1 - 2.92e8iT - 5.96e16T^{2} \)
7 \( 1 - 2.55e10iT - 1.91e20T^{2} \)
11 \( 1 - 3.79e12iT - 9.84e24T^{2} \)
13 \( 1 - 1.53e12T + 5.42e26T^{2} \)
17 \( 1 + 5.00e14iT - 3.39e29T^{2} \)
19 \( 1 + 2.76e14iT - 4.89e30T^{2} \)
29 \( 1 + 4.83e17T + 1.25e35T^{2} \)
31 \( 1 - 7.59e17T + 6.20e35T^{2} \)
37 \( 1 - 7.18e18iT - 4.33e37T^{2} \)
41 \( 1 - 1.24e19T + 5.09e38T^{2} \)
43 \( 1 - 1.31e18iT - 1.59e39T^{2} \)
47 \( 1 - 1.07e20T + 1.35e40T^{2} \)
53 \( 1 + 6.62e20iT - 2.41e41T^{2} \)
59 \( 1 - 5.97e20T + 3.16e42T^{2} \)
61 \( 1 - 1.57e21iT - 7.04e42T^{2} \)
67 \( 1 - 1.37e22iT - 6.69e43T^{2} \)
71 \( 1 + 1.82e22T + 2.69e44T^{2} \)
73 \( 1 + 9.66e21T + 5.24e44T^{2} \)
79 \( 1 + 7.92e22iT - 3.49e45T^{2} \)
83 \( 1 - 5.83e22iT - 1.14e46T^{2} \)
89 \( 1 + 3.41e23iT - 6.10e46T^{2} \)
97 \( 1 - 2.03e23iT - 4.81e47T^{2} \)
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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

−12.87580137136027075080506418235, −11.85150960848868653144926340682, −11.55842743848677405110872501405, −10.20652062859951996599503380766, −7.14825712650312882916988740018, −6.24770296439985562426360068723, −5.52236516836952741322207551803, −4.54911509876922103970764010800, −2.86529713667835952238624655309, −2.02197327137592802765582007436, 0.54228458820311502659541460970, 1.40385664735836214828015175959, 3.75955419679176076208649177181, 4.41409207492032236744374908283, 5.52256306228524440107357084477, 6.27114767220233752649136704015, 7.62092919186443431687459329598, 10.54312499830223779555860187665, 11.22406660021545273904573750863, 12.35656876109595911674059090612

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