↑

Properties

Label 2-23-23.22-c24-0-25
Degree $2$
Conductor $23$
Sign $-0.180 + 0.983i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.53e3·2-s − 9.24e5·3-s + 3.82e6·4-s − 1.58e8i·5-s − 4.19e9·6-s + 1.60e10i·7-s − 5.87e10·8-s + 5.71e11·9-s − 7.21e11i·10-s − 2.13e12i·11-s − 3.53e12·12-s − 1.48e13·13-s + 7.26e13i·14-s + 1.46e14i·15-s − 3.31e14·16-s + 7.71e14i·17-s + ⋯
L(s)  = 1  + 1.10·2-s − 1.73·3-s + 0.228·4-s − 0.651i·5-s − 1.92·6-s + 1.15i·7-s − 0.855·8-s + 2.02·9-s − 0.721i·10-s − 0.681i·11-s − 0.396·12-s − 0.637·13-s + 1.28i·14-s + 1.13i·15-s − 1.17·16-s + 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.180 + 0.983i$
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: \((2,\ 23,\ (\ :12),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.4134343617\)
\(L(\frac12)\) \(\approx\) \(0.4134343617\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (3.95e15 - 2.15e16i)T \)
good2 \( 1 - 4.53e3T + 1.67e7T^{2} \)
3 \( 1 + 9.24e5T + 2.82e11T^{2} \)
5 \( 1 + 1.58e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.60e10iT - 1.91e20T^{2} \)
11 \( 1 + 2.13e12iT - 9.84e24T^{2} \)
13 \( 1 + 1.48e13T + 5.42e26T^{2} \)
17 \( 1 - 7.71e14iT - 3.39e29T^{2} \)
19 \( 1 - 3.82e15iT - 4.89e30T^{2} \)
29 \( 1 - 3.40e17T + 1.25e35T^{2} \)
31 \( 1 + 8.12e16T + 6.20e35T^{2} \)
37 \( 1 + 1.91e18iT - 4.33e37T^{2} \)
41 \( 1 + 3.29e19T + 5.09e38T^{2} \)
43 \( 1 + 5.82e19iT - 1.59e39T^{2} \)
47 \( 1 + 2.17e20T + 1.35e40T^{2} \)
53 \( 1 + 7.16e20iT - 2.41e41T^{2} \)
59 \( 1 - 5.02e19T + 3.16e42T^{2} \)
61 \( 1 - 8.81e20iT - 7.04e42T^{2} \)
67 \( 1 - 7.09e21iT - 6.69e43T^{2} \)
71 \( 1 + 2.01e22T + 2.69e44T^{2} \)
73 \( 1 - 2.39e22T + 5.24e44T^{2} \)
79 \( 1 + 9.56e22iT - 3.49e45T^{2} \)
83 \( 1 + 1.36e22iT - 1.14e46T^{2} \)
89 \( 1 - 1.22e23iT - 6.10e46T^{2} \)
97 \( 1 + 6.75e23iT - 4.81e47T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.22818761850603271972747994869, −11.74862964631641768905255598725, −10.15658441603128974551101742220, −8.530449184359905476810059922103, −6.34288638087616839441316456837, −5.59196275018037350997186387293, −4.98632403344081087750573805087, −3.64148620952373615596071561856, −1.63985752547020618826684798139, −0.11992159262414722321726868181, 0.75130269602377993089859429031, 2.86965674660998243268285075678, 4.66920662474999169778621569087, 4.81587232582595169614848312536, 6.56510124297918340606122981071, 7.02234825421114523055127787552, 9.803742724265083894664626776985, 10.92265388399768114309863560330, 11.83938523755962945860275579201, 12.89371346321137469630176419380

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