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Properties

Label 2-23-23.22-c12-0-19
Degree $2$
Conductor $23$
Sign $0.444 + 0.895i$
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  + 68.2·2-s + 1.04e3·3-s + 564.·4-s − 1.03e4i·5-s + 7.12e4·6-s − 1.48e5i·7-s − 2.41e5·8-s + 5.56e5·9-s − 7.09e5i·10-s − 2.82e6i·11-s + 5.89e5·12-s + 7.25e6·13-s − 1.01e7i·14-s − 1.08e7i·15-s − 1.87e7·16-s + 4.15e7i·17-s + ⋯
L(s)  = 1  + 1.06·2-s + 1.43·3-s + 0.137·4-s − 0.665i·5-s + 1.52·6-s − 1.26i·7-s − 0.919·8-s + 1.04·9-s − 0.709i·10-s − 1.59i·11-s + 0.197·12-s + 1.50·13-s − 1.34i·14-s − 0.951i·15-s − 1.11·16-s + 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.444 + 0.895i$
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: \((2,\ 23,\ (\ :6),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(3.98968 - 2.47414i\)
\(L(\frac12)\) \(\approx\) \(3.98968 - 2.47414i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-6.58e7 - 1.32e8i)T \)
good2 \( 1 - 68.2T + 4.09e3T^{2} \)
3 \( 1 - 1.04e3T + 5.31e5T^{2} \)
5 \( 1 + 1.03e4iT - 2.44e8T^{2} \)
7 \( 1 + 1.48e5iT - 1.38e10T^{2} \)
11 \( 1 + 2.82e6iT - 3.13e12T^{2} \)
13 \( 1 - 7.25e6T + 2.32e13T^{2} \)
17 \( 1 - 4.15e7iT - 5.82e14T^{2} \)
19 \( 1 - 1.46e7iT - 2.21e15T^{2} \)
29 \( 1 + 5.60e8T + 3.53e17T^{2} \)
31 \( 1 - 1.23e9T + 7.87e17T^{2} \)
37 \( 1 + 4.16e9iT - 6.58e18T^{2} \)
41 \( 1 + 2.24e9T + 2.25e19T^{2} \)
43 \( 1 - 3.28e9iT - 3.99e19T^{2} \)
47 \( 1 + 3.04e9T + 1.16e20T^{2} \)
53 \( 1 - 3.72e10iT - 4.91e20T^{2} \)
59 \( 1 + 2.34e10T + 1.77e21T^{2} \)
61 \( 1 + 2.00e10iT - 2.65e21T^{2} \)
67 \( 1 - 5.90e10iT - 8.18e21T^{2} \)
71 \( 1 - 1.87e11T + 1.64e22T^{2} \)
73 \( 1 + 8.56e9T + 2.29e22T^{2} \)
79 \( 1 + 3.28e10iT - 5.90e22T^{2} \)
83 \( 1 - 2.20e11iT - 1.06e23T^{2} \)
89 \( 1 + 3.96e11iT - 2.46e23T^{2} \)
97 \( 1 - 1.87e11iT - 6.93e23T^{2} \)
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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

−14.29860624613873245047612604286, −13.55744932917762807664462271802, −13.00882518275746913914807722174, −10.92358226994305433859085441295, −8.977505653967855851354795295333, −8.154325899220229014462050641783, −5.97870426752669851711886041594, −3.96965947076513835326341700153, −3.41396585453076705221641459189, −1.11883229294389800084834971874, 2.36419605175926772715549047307, 3.21352403409267193447187989564, 4.84298166151563940691135903588, 6.70252836171352548125559538418, 8.554994359409990582476959556426, 9.546705713684497343585723660238, 11.75515433154099510643144534933, 13.05761408811753559998856808123, 14.02935858846595481461189165149, 15.05107341127182469119820656405

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