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Properties

Label 2-23-23.22-c24-0-11
Degree $2$
Conductor $23$
Sign $0.740 - 0.672i$
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  − 2.73e3·2-s − 1.09e4·3-s − 9.27e6·4-s + 2.86e8i·5-s + 3.00e7·6-s − 2.46e10i·7-s + 7.13e10·8-s − 2.82e11·9-s − 7.85e11i·10-s + 4.90e12i·11-s + 1.01e11·12-s − 1.92e13·13-s + 6.74e13i·14-s − 3.14e12i·15-s − 3.99e13·16-s + 1.36e14i·17-s + ⋯
L(s)  = 1  − 0.668·2-s − 0.0206·3-s − 0.552·4-s + 1.17i·5-s + 0.0137·6-s − 1.77i·7-s + 1.03·8-s − 0.999·9-s − 0.785i·10-s + 1.56i·11-s + 0.0113·12-s − 0.827·13-s + 1.18i·14-s − 0.0242i·15-s − 0.141·16-s + 0.233i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.740 - 0.672i$
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: \((2,\ 23,\ (\ :12),\ 0.740 - 0.672i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.5548420788\)
\(L(\frac12)\) \(\approx\) \(0.5548420788\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-1.62e16 + 1.47e16i)T \)
good2 \( 1 + 2.73e3T + 1.67e7T^{2} \)
3 \( 1 + 1.09e4T + 2.82e11T^{2} \)
5 \( 1 - 2.86e8iT - 5.96e16T^{2} \)
7 \( 1 + 2.46e10iT - 1.91e20T^{2} \)
11 \( 1 - 4.90e12iT - 9.84e24T^{2} \)
13 \( 1 + 1.92e13T + 5.42e26T^{2} \)
17 \( 1 - 1.36e14iT - 3.39e29T^{2} \)
19 \( 1 + 2.47e15iT - 4.89e30T^{2} \)
29 \( 1 + 6.01e17T + 1.25e35T^{2} \)
31 \( 1 + 5.44e17T + 6.20e35T^{2} \)
37 \( 1 + 1.13e19iT - 4.33e37T^{2} \)
41 \( 1 + 3.48e19T + 5.09e38T^{2} \)
43 \( 1 + 8.67e18iT - 1.59e39T^{2} \)
47 \( 1 - 1.32e20T + 1.35e40T^{2} \)
53 \( 1 - 1.82e20iT - 2.41e41T^{2} \)
59 \( 1 + 7.47e20T + 3.16e42T^{2} \)
61 \( 1 + 7.14e20iT - 7.04e42T^{2} \)
67 \( 1 - 4.33e21iT - 6.69e43T^{2} \)
71 \( 1 - 2.99e22T + 2.69e44T^{2} \)
73 \( 1 + 1.89e22T + 5.24e44T^{2} \)
79 \( 1 - 1.64e22iT - 3.49e45T^{2} \)
83 \( 1 + 3.52e22iT - 1.14e46T^{2} \)
89 \( 1 + 1.37e23iT - 6.10e46T^{2} \)
97 \( 1 + 2.08e23iT - 4.81e47T^{2} \)
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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

−12.96467146346140962817241319662, −11.00761470505394700445292219014, −10.34511643932101632021620516243, −9.237574516499895911784659695328, −7.43456200660055703596018902242, −7.07537883000380991633449899187, −4.88039624421992169308345830987, −3.71515669392206649871304860156, −2.15489973283334362907287567397, −0.50898251573542446443494791007, 0.33334107163736145607717842801, 1.66993811606580762549508773074, 3.23222530983353507799883564913, 5.17458493335211665322912777538, 5.65983371934544803752309408875, 8.154797038919837650680494452489, 8.758397863520932630997009211891, 9.461810656651725486723737112799, 11.36424130419247108234472217920, 12.41256870232541937989133729720

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