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Properties

Label 2-23-23.22-c24-0-35
Degree $2$
Conductor $23$
Sign $0.985 + 0.170i$
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  + 2.54e3·2-s + 6.99e5·3-s − 1.03e7·4-s + 3.93e8i·5-s + 1.77e9·6-s − 1.74e10i·7-s − 6.88e10·8-s + 2.06e11·9-s + 1.00e12i·10-s − 1.74e12i·11-s − 7.21e12·12-s + 1.93e13·13-s − 4.42e13i·14-s + 2.75e14i·15-s − 1.98e12·16-s − 4.91e14i·17-s + ⋯
L(s)  = 1  + 0.620·2-s + 1.31·3-s − 0.614·4-s + 1.61i·5-s + 0.816·6-s − 1.25i·7-s − 1.00·8-s + 0.731·9-s + 1.00i·10-s − 0.556i·11-s − 0.809·12-s + 0.832·13-s − 0.781i·14-s + 2.12i·15-s − 0.00705·16-s − 0.844i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.985 + 0.170i$
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: \((2,\ 23,\ (\ :12),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(4.058688026\)
\(L(\frac12)\) \(\approx\) \(4.058688026\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-2.15e16 - 3.74e15i)T \)
good2 \( 1 - 2.54e3T + 1.67e7T^{2} \)
3 \( 1 - 6.99e5T + 2.82e11T^{2} \)
5 \( 1 - 3.93e8iT - 5.96e16T^{2} \)
7 \( 1 + 1.74e10iT - 1.91e20T^{2} \)
11 \( 1 + 1.74e12iT - 9.84e24T^{2} \)
13 \( 1 - 1.93e13T + 5.42e26T^{2} \)
17 \( 1 + 4.91e14iT - 3.39e29T^{2} \)
19 \( 1 + 1.35e15iT - 4.89e30T^{2} \)
29 \( 1 - 5.62e17T + 1.25e35T^{2} \)
31 \( 1 - 6.23e17T + 6.20e35T^{2} \)
37 \( 1 - 2.82e18iT - 4.33e37T^{2} \)
41 \( 1 - 1.28e19T + 5.09e38T^{2} \)
43 \( 1 - 3.38e19iT - 1.59e39T^{2} \)
47 \( 1 - 4.15e19T + 1.35e40T^{2} \)
53 \( 1 - 4.49e20iT - 2.41e41T^{2} \)
59 \( 1 - 1.91e20T + 3.16e42T^{2} \)
61 \( 1 + 1.43e21iT - 7.04e42T^{2} \)
67 \( 1 - 3.75e21iT - 6.69e43T^{2} \)
71 \( 1 - 1.87e22T + 2.69e44T^{2} \)
73 \( 1 + 2.86e22T + 5.24e44T^{2} \)
79 \( 1 + 1.13e23iT - 3.49e45T^{2} \)
83 \( 1 + 1.35e23iT - 1.14e46T^{2} \)
89 \( 1 - 1.69e23iT - 6.10e46T^{2} \)
97 \( 1 + 5.75e23iT - 4.81e47T^{2} \)
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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

−13.42515738342064862932117782183, −11.27320567404834273393727483182, −10.10877192696855833870755847584, −8.804843578297079796487914118842, −7.52241568583110935224109784595, −6.38310191807294332600673929573, −4.39838461274891466136340714942, −3.23401376781882517524722037821, −2.88537419747965643569332617680, −0.77811643127433928114656650260, 1.04088345588801947963689247501, 2.36529870831182477186023867658, 3.69976992858822874436828406656, 4.76872747250618900906585111748, 5.84492945664984340082988694753, 8.480290623322781179993831859648, 8.557367012382314007652894794713, 9.597544531596667580270256151672, 12.18476967214366148786775631284, 12.81228623518846271065406331321

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