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Properties

Label 2-23-23.22-c24-0-13
Degree $2$
Conductor $23$
Sign $-0.610 - 0.792i$
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  − 263.·2-s − 5.64e5·3-s − 1.67e7·4-s − 5.52e7i·5-s + 1.48e8·6-s + 2.02e10i·7-s + 8.83e9·8-s + 3.60e10·9-s + 1.45e10i·10-s + 3.02e12i·11-s + 9.42e12·12-s − 5.50e12·13-s − 5.33e12i·14-s + 3.11e13i·15-s + 2.77e14·16-s + 4.43e14i·17-s + ⋯
L(s)  = 1  − 0.0644·2-s − 1.06·3-s − 0.995·4-s − 0.226i·5-s + 0.0683·6-s + 1.46i·7-s + 0.128·8-s + 0.127·9-s + 0.0145i·10-s + 0.963i·11-s + 1.05·12-s − 0.236·13-s − 0.0941i·14-s + 0.240i·15-s + 0.987·16-s + 0.761i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.610 - 0.792i$
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: \((2,\ 23,\ (\ :12),\ -0.610 - 0.792i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.7713941592\)
\(L(\frac12)\) \(\approx\) \(0.7713941592\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.33e16 + 1.73e16i)T \)
good2 \( 1 + 263.T + 1.67e7T^{2} \)
3 \( 1 + 5.64e5T + 2.82e11T^{2} \)
5 \( 1 + 5.52e7iT - 5.96e16T^{2} \)
7 \( 1 - 2.02e10iT - 1.91e20T^{2} \)
11 \( 1 - 3.02e12iT - 9.84e24T^{2} \)
13 \( 1 + 5.50e12T + 5.42e26T^{2} \)
17 \( 1 - 4.43e14iT - 3.39e29T^{2} \)
19 \( 1 + 1.32e15iT - 4.89e30T^{2} \)
29 \( 1 - 6.24e17T + 1.25e35T^{2} \)
31 \( 1 + 2.11e17T + 6.20e35T^{2} \)
37 \( 1 + 3.46e18iT - 4.33e37T^{2} \)
41 \( 1 + 7.55e17T + 5.09e38T^{2} \)
43 \( 1 - 5.05e19iT - 1.59e39T^{2} \)
47 \( 1 - 1.75e20T + 1.35e40T^{2} \)
53 \( 1 - 6.08e20iT - 2.41e41T^{2} \)
59 \( 1 - 8.70e18T + 3.16e42T^{2} \)
61 \( 1 + 8.24e20iT - 7.04e42T^{2} \)
67 \( 1 + 6.73e21iT - 6.69e43T^{2} \)
71 \( 1 - 2.20e22T + 2.69e44T^{2} \)
73 \( 1 - 1.30e22T + 5.24e44T^{2} \)
79 \( 1 - 4.27e22iT - 3.49e45T^{2} \)
83 \( 1 - 6.81e22iT - 1.14e46T^{2} \)
89 \( 1 - 3.63e23iT - 6.10e46T^{2} \)
97 \( 1 + 9.98e23iT - 4.81e47T^{2} \)
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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

−12.55251539738003871540439688493, −12.22870717200442557106731713414, −10.61531731421869621544920664115, −9.295699145281006987055842823541, −8.319069428363504263785724629112, −6.37600690384006840871370731674, −5.28762034333780842769241941063, −4.50559301340555539382004440696, −2.52365217641088944391523177197, −0.874290591211453060635721925415, 0.38303595534458422996581558820, 0.926686998774859902737590678281, 3.35980103562382760912200478349, 4.55607463624727102300591779069, 5.65231259671654163258777043625, 7.01173428477769062300236507081, 8.414435081752463642122161096170, 10.02410463065544165796517643595, 10.84901566374579934894403159041, 12.11983604240283073637940517742

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