Properties

Label 2-23-23.22-c14-0-7
Degree $2$
Conductor $23$
Sign $0.00196 - 0.999i$
Analytic cond. $28.5956$
Root an. cond. $5.34749$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 176.·2-s − 1.55e3·3-s + 1.48e4·4-s + 2.86e4i·5-s + 2.74e5·6-s + 1.02e6i·7-s + 2.66e5·8-s − 2.36e6·9-s − 5.05e6i·10-s − 7.94e6i·11-s − 2.31e7·12-s + 7.50e7·13-s − 1.80e8i·14-s − 4.45e7i·15-s − 2.90e8·16-s − 3.73e8i·17-s + ⋯
L(s)  = 1  − 1.38·2-s − 0.711·3-s + 0.907·4-s + 0.366i·5-s + 0.982·6-s + 1.24i·7-s + 0.127·8-s − 0.494·9-s − 0.505i·10-s − 0.407i·11-s − 0.645·12-s + 1.19·13-s − 1.71i·14-s − 0.260i·15-s − 1.08·16-s − 0.910i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00196 - 0.999i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.00196 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.00196 - 0.999i$
Analytic conductor: \(28.5956\)
Root analytic conductor: \(5.34749\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :7),\ 0.00196 - 0.999i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.5201899898\)
\(L(\frac12)\) \(\approx\) \(0.5201899898\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (6.69e6 - 3.40e9i)T \)
good2 \( 1 + 176.T + 1.63e4T^{2} \)
3 \( 1 + 1.55e3T + 4.78e6T^{2} \)
5 \( 1 - 2.86e4iT - 6.10e9T^{2} \)
7 \( 1 - 1.02e6iT - 6.78e11T^{2} \)
11 \( 1 + 7.94e6iT - 3.79e14T^{2} \)
13 \( 1 - 7.50e7T + 3.93e15T^{2} \)
17 \( 1 + 3.73e8iT - 1.68e17T^{2} \)
19 \( 1 + 1.47e9iT - 7.99e17T^{2} \)
29 \( 1 + 2.04e10T + 2.97e20T^{2} \)
31 \( 1 - 1.05e10T + 7.56e20T^{2} \)
37 \( 1 - 1.71e11iT - 9.01e21T^{2} \)
41 \( 1 - 1.96e11T + 3.79e22T^{2} \)
43 \( 1 + 1.63e11iT - 7.38e22T^{2} \)
47 \( 1 - 1.29e11T + 2.56e23T^{2} \)
53 \( 1 + 1.49e12iT - 1.37e24T^{2} \)
59 \( 1 - 2.16e11T + 6.19e24T^{2} \)
61 \( 1 - 1.53e12iT - 9.87e24T^{2} \)
67 \( 1 - 2.75e12iT - 3.67e25T^{2} \)
71 \( 1 - 8.43e12T + 8.27e25T^{2} \)
73 \( 1 + 1.34e13T + 1.22e26T^{2} \)
79 \( 1 - 3.08e13iT - 3.68e26T^{2} \)
83 \( 1 - 1.79e13iT - 7.36e26T^{2} \)
89 \( 1 - 2.82e13iT - 1.95e27T^{2} \)
97 \( 1 - 1.46e14iT - 6.52e27T^{2} \)
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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

−15.38497146528021665624884681510, −13.52349296776270233054029278218, −11.56281758810526041042810962720, −11.02238730642551709206488743449, −9.322131215248779472031457894536, −8.470045787149720590509823160076, −6.74150931349793108355562859974, −5.35085317907718175850054861385, −2.70193970106383696577912370832, −0.891483073013022757782725018301, 0.42985690112440866944188395007, 1.45029874275154884542109969068, 4.13253692898809722295574821531, 6.11406876745030788875703685020, 7.65275150143523508273407972320, 8.814075772481818182985054441673, 10.38993089796251741115018601020, 10.98711670801094566409577184410, 12.71533129818285643637296299111, 14.26777917957455041788809401628

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