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Properties

Label 2-23-23.22-c24-0-2
Degree $2$
Conductor $23$
Sign $-0.983 + 0.178i$
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  − 5.48e3·2-s + 2.36e5·3-s + 1.33e7·4-s − 1.99e8i·5-s − 1.29e9·6-s + 1.19e10i·7-s + 1.88e10·8-s − 2.26e11·9-s + 1.09e12i·10-s + 5.35e12i·11-s + 3.15e12·12-s + 2.10e13·13-s − 6.56e13i·14-s − 4.71e13i·15-s − 3.27e14·16-s + 6.91e14i·17-s + ⋯
L(s)  = 1  − 1.34·2-s + 0.444·3-s + 0.795·4-s − 0.817i·5-s − 0.596·6-s + 0.863i·7-s + 0.273·8-s − 0.802·9-s + 1.09i·10-s + 1.70i·11-s + 0.353·12-s + 0.903·13-s − 1.15i·14-s − 0.363i·15-s − 1.16·16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.983 + 0.178i$
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: \((2,\ 23,\ (\ :12),\ -0.983 + 0.178i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.2614294596\)
\(L(\frac12)\) \(\approx\) \(0.2614294596\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (2.15e16 - 3.91e15i)T \)
good2 \( 1 + 5.48e3T + 1.67e7T^{2} \)
3 \( 1 - 2.36e5T + 2.82e11T^{2} \)
5 \( 1 + 1.99e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.19e10iT - 1.91e20T^{2} \)
11 \( 1 - 5.35e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.10e13T + 5.42e26T^{2} \)
17 \( 1 - 6.91e14iT - 3.39e29T^{2} \)
19 \( 1 + 3.99e14iT - 4.89e30T^{2} \)
29 \( 1 + 1.94e17T + 1.25e35T^{2} \)
31 \( 1 + 5.00e17T + 6.20e35T^{2} \)
37 \( 1 - 3.15e18iT - 4.33e37T^{2} \)
41 \( 1 - 2.60e19T + 5.09e38T^{2} \)
43 \( 1 + 3.07e19iT - 1.59e39T^{2} \)
47 \( 1 - 1.71e20T + 1.35e40T^{2} \)
53 \( 1 + 2.57e19iT - 2.41e41T^{2} \)
59 \( 1 + 6.69e20T + 3.16e42T^{2} \)
61 \( 1 - 2.38e21iT - 7.04e42T^{2} \)
67 \( 1 - 9.93e21iT - 6.69e43T^{2} \)
71 \( 1 + 5.93e21T + 2.69e44T^{2} \)
73 \( 1 + 4.39e22T + 5.24e44T^{2} \)
79 \( 1 + 1.11e23iT - 3.49e45T^{2} \)
83 \( 1 + 8.85e22iT - 1.14e46T^{2} \)
89 \( 1 + 1.55e23iT - 6.10e46T^{2} \)
97 \( 1 - 6.49e23iT - 4.81e47T^{2} \)
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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

−13.12840514952340237453192544839, −11.94252332821566218085359259859, −10.43106049429218584211922951176, −9.129867131708514128861455566511, −8.668514075106340411483004530729, −7.54873188596352812160488536044, −5.78245454554967627700020738709, −4.22100953792282041105657603216, −2.26585875603035488828368683584, −1.42099607233263167136582263406, 0.10664429598255322493588679209, 0.988785017535939320711917587121, 2.62646224568950785475594357682, 3.74577796478785971967389155523, 5.97555755749788709358469322992, 7.35738465292709177667376836557, 8.342261333683765376331627244714, 9.308107520307298684293148190798, 10.77226506534951339580171402060, 11.20224846439132992429473664437

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