Properties

Label 2-23-23.22-c12-0-15
Degree $2$
Conductor $23$
Sign $-0.823 - 0.567i$
Analytic cond. $21.0218$
Root an. cond. $4.58495$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 43.1·2-s − 1.08e3·3-s − 2.23e3·4-s − 197. i·5-s + 4.67e4·6-s − 1.38e5i·7-s + 2.73e5·8-s + 6.42e5·9-s + 8.50e3i·10-s − 3.09e6i·11-s + 2.41e6·12-s − 7.82e5·13-s + 5.98e6i·14-s + 2.13e5i·15-s − 2.64e6·16-s + 1.68e7i·17-s + ⋯
L(s)  = 1  − 0.674·2-s − 1.48·3-s − 0.545·4-s − 0.0126i·5-s + 1.00·6-s − 1.17i·7-s + 1.04·8-s + 1.20·9-s + 0.00850i·10-s − 1.74i·11-s + 0.809·12-s − 0.162·13-s + 0.795i·14-s + 0.0187i·15-s − 0.157·16-s + 0.696i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(21.0218\)
Root analytic conductor: \(4.58495\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :6),\ -0.823 - 0.567i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0795238 + 0.255298i\)
\(L(\frac12)\) \(\approx\) \(0.0795238 + 0.255298i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.21e8 + 8.40e7i)T \)
good2 \( 1 + 43.1T + 4.09e3T^{2} \)
3 \( 1 + 1.08e3T + 5.31e5T^{2} \)
5 \( 1 + 197. iT - 2.44e8T^{2} \)
7 \( 1 + 1.38e5iT - 1.38e10T^{2} \)
11 \( 1 + 3.09e6iT - 3.13e12T^{2} \)
13 \( 1 + 7.82e5T + 2.32e13T^{2} \)
17 \( 1 - 1.68e7iT - 5.82e14T^{2} \)
19 \( 1 + 8.31e7iT - 2.21e15T^{2} \)
29 \( 1 + 4.26e8T + 3.53e17T^{2} \)
31 \( 1 + 1.02e9T + 7.87e17T^{2} \)
37 \( 1 + 3.34e9iT - 6.58e18T^{2} \)
41 \( 1 - 3.61e9T + 2.25e19T^{2} \)
43 \( 1 + 7.08e9iT - 3.99e19T^{2} \)
47 \( 1 - 1.41e10T + 1.16e20T^{2} \)
53 \( 1 + 7.07e9iT - 4.91e20T^{2} \)
59 \( 1 + 3.08e10T + 1.77e21T^{2} \)
61 \( 1 + 1.27e9iT - 2.65e21T^{2} \)
67 \( 1 - 5.88e10iT - 8.18e21T^{2} \)
71 \( 1 + 7.96e10T + 1.64e22T^{2} \)
73 \( 1 + 2.01e11T + 2.29e22T^{2} \)
79 \( 1 - 2.84e11iT - 5.90e22T^{2} \)
83 \( 1 - 3.50e11iT - 1.06e23T^{2} \)
89 \( 1 + 1.59e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.07e11iT - 6.93e23T^{2} \)
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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

−14.01279545036515582288446526562, −12.89671255270353844924558202410, −11.03240935358561795069046316255, −10.62035016704254068720706736788, −8.854766570359795873469064105633, −7.17133547711513113807116672444, −5.61017935377370744502011599374, −4.13776561662642958024890286810, −0.807741249077975678173008916831, −0.23367460959219318803583542349, 1.61112067507566336827242792741, 4.63364572246029743185988383606, 5.76017226218434878640405145631, 7.51382096422580486230978369868, 9.324507153734771350491323891112, 10.33946555407917151048993425009, 11.91209627051562005952101900034, 12.66657969234140617185223707557, 14.70952211516191774437466249705, 16.18289866415556656777530077251

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