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Properties

Label 2-23-23.22-c12-0-22
Degree $2$
Conductor $23$
Sign $-0.980 - 0.195i$
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  + 20.5·2-s + 729.·3-s − 3.67e3·4-s − 1.95e4i·5-s + 1.50e4·6-s − 5.51e3i·7-s − 1.59e5·8-s + 1.05e3·9-s − 4.01e5i·10-s + 2.84e6i·11-s − 2.68e6·12-s − 7.37e6·13-s − 1.13e5i·14-s − 1.42e7i·15-s + 1.17e7·16-s + 2.30e7i·17-s + ⋯
L(s)  = 1  + 0.321·2-s + 1.00·3-s − 0.896·4-s − 1.25i·5-s + 0.321·6-s − 0.0468i·7-s − 0.609·8-s + 0.00198·9-s − 0.401i·10-s + 1.60i·11-s − 0.897·12-s − 1.52·13-s − 0.0150i·14-s − 1.25i·15-s + 0.701·16-s + 0.954i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.980 - 0.195i$
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: \((2,\ 23,\ (\ :6),\ -0.980 - 0.195i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0206360 + 0.209177i\)
\(L(\frac12)\) \(\approx\) \(0.0206360 + 0.209177i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.45e8 + 2.89e7i)T \)
good2 \( 1 - 20.5T + 4.09e3T^{2} \)
3 \( 1 - 729.T + 5.31e5T^{2} \)
5 \( 1 + 1.95e4iT - 2.44e8T^{2} \)
7 \( 1 + 5.51e3iT - 1.38e10T^{2} \)
11 \( 1 - 2.84e6iT - 3.13e12T^{2} \)
13 \( 1 + 7.37e6T + 2.32e13T^{2} \)
17 \( 1 - 2.30e7iT - 5.82e14T^{2} \)
19 \( 1 + 5.27e7iT - 2.21e15T^{2} \)
29 \( 1 + 2.08e8T + 3.53e17T^{2} \)
31 \( 1 + 9.37e8T + 7.87e17T^{2} \)
37 \( 1 + 4.16e9iT - 6.58e18T^{2} \)
41 \( 1 + 2.43e9T + 2.25e19T^{2} \)
43 \( 1 - 1.05e10iT - 3.99e19T^{2} \)
47 \( 1 - 9.05e9T + 1.16e20T^{2} \)
53 \( 1 + 3.52e10iT - 4.91e20T^{2} \)
59 \( 1 + 1.98e10T + 1.77e21T^{2} \)
61 \( 1 + 9.34e9iT - 2.65e21T^{2} \)
67 \( 1 + 1.05e11iT - 8.18e21T^{2} \)
71 \( 1 + 2.74e10T + 1.64e22T^{2} \)
73 \( 1 - 4.00e10T + 2.29e22T^{2} \)
79 \( 1 - 1.55e11iT - 5.90e22T^{2} \)
83 \( 1 - 2.09e11iT - 1.06e23T^{2} \)
89 \( 1 - 1.16e10iT - 2.46e23T^{2} \)
97 \( 1 + 1.10e12iT - 6.93e23T^{2} \)
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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

−14.36078787746753385713487110038, −12.96914779653272991139543811738, −12.37751295997270632469960173248, −9.707842762317721553689347889824, −8.971154020716816892844286747572, −7.69965293651804164448103151124, −5.13647578969611597397364903251, −4.12364195351228701784064108050, −2.12715977783106452894137994039, −0.05637885266979393845042584331, 2.68767578151470001111662526332, 3.61917040778296613207315011977, 5.68260637328032504058622159435, 7.60553042654306510455595309670, 8.906269077613372748432909904250, 10.19525130815638766490447081611, 11.90178573817477746024821931049, 13.76135828120185083296546901335, 14.17613968941915136822142924884, 15.08563330279133427672548477383

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