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Properties

Label 2-23-23.22-c24-0-6
Degree $2$
Conductor $23$
Sign $0.191 - 0.981i$
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  + 2.98e3·2-s + 1.15e5·3-s − 7.86e6·4-s − 1.34e8i·5-s + 3.44e8·6-s − 1.30e10i·7-s − 7.35e10·8-s − 2.69e11·9-s − 4.01e11i·10-s − 2.85e12i·11-s − 9.08e11·12-s − 2.36e13·13-s − 3.90e13i·14-s − 1.55e13i·15-s − 8.77e13·16-s + 4.84e14i·17-s + ⋯
L(s)  = 1  + 0.728·2-s + 0.217·3-s − 0.468·4-s − 0.550i·5-s + 0.158·6-s − 0.945i·7-s − 1.07·8-s − 0.952·9-s − 0.401i·10-s − 0.909i·11-s − 0.101·12-s − 1.01·13-s − 0.689i·14-s − 0.119i·15-s − 0.311·16-s + 0.831i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.191 - 0.981i$
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: \((2,\ 23,\ (\ :12),\ 0.191 - 0.981i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.6824021813\)
\(L(\frac12)\) \(\approx\) \(0.6824021813\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-4.19e15 + 2.15e16i)T \)
good2 \( 1 - 2.98e3T + 1.67e7T^{2} \)
3 \( 1 - 1.15e5T + 2.82e11T^{2} \)
5 \( 1 + 1.34e8iT - 5.96e16T^{2} \)
7 \( 1 + 1.30e10iT - 1.91e20T^{2} \)
11 \( 1 + 2.85e12iT - 9.84e24T^{2} \)
13 \( 1 + 2.36e13T + 5.42e26T^{2} \)
17 \( 1 - 4.84e14iT - 3.39e29T^{2} \)
19 \( 1 - 1.60e15iT - 4.89e30T^{2} \)
29 \( 1 + 9.26e16T + 1.25e35T^{2} \)
31 \( 1 - 4.37e17T + 6.20e35T^{2} \)
37 \( 1 - 1.54e18iT - 4.33e37T^{2} \)
41 \( 1 + 7.65e18T + 5.09e38T^{2} \)
43 \( 1 - 2.83e19iT - 1.59e39T^{2} \)
47 \( 1 - 4.11e19T + 1.35e40T^{2} \)
53 \( 1 - 3.21e20iT - 2.41e41T^{2} \)
59 \( 1 + 5.45e20T + 3.16e42T^{2} \)
61 \( 1 - 2.60e20iT - 7.04e42T^{2} \)
67 \( 1 + 1.94e21iT - 6.69e43T^{2} \)
71 \( 1 + 1.51e22T + 2.69e44T^{2} \)
73 \( 1 + 1.69e21T + 5.24e44T^{2} \)
79 \( 1 + 2.50e22iT - 3.49e45T^{2} \)
83 \( 1 - 2.83e22iT - 1.14e46T^{2} \)
89 \( 1 - 7.63e22iT - 6.10e46T^{2} \)
97 \( 1 - 7.48e23iT - 4.81e47T^{2} \)
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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

−13.09465240161598194027223363294, −12.02571000038760639363063881703, −10.49492019041374786884879395008, −9.012402923201818034466664784031, −8.052075494381319973682646289569, −6.21194198785295955515092268311, −5.03232424063229218128559259396, −3.94837746344953421394154133505, −2.82270291992244215168163614039, −0.886311068148860451835297547655, 0.14595803164210824714332589469, 2.38072771319292203364955249750, 3.11711724228529261247112193745, 4.75132161488533388279730062466, 5.62662372518212161033834627532, 7.14202092796205768752299911062, 8.774530477326998647023000579298, 9.687069646732790312064098937128, 11.53947176461064779278673902183, 12.42781593885342769103884022629

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