Properties

Label 2-23-1.1-c15-0-16
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $32.8195$
Root an. cond. $5.72883$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.5·2-s − 2.38e3·3-s − 3.25e4·4-s + 1.72e5·5-s + 3.69e4·6-s − 5.70e5·7-s + 1.01e6·8-s − 8.67e6·9-s − 2.68e6·10-s + 7.47e7·11-s + 7.74e7·12-s + 2.00e8·13-s + 8.85e6·14-s − 4.11e8·15-s + 1.05e9·16-s + 1.04e9·17-s + 1.34e8·18-s − 4.82e9·19-s − 5.62e9·20-s + 1.35e9·21-s − 1.16e9·22-s + 3.40e9·23-s − 2.41e9·24-s − 6.11e8·25-s − 3.10e9·26-s + 5.48e10·27-s + 1.85e10·28-s + ⋯
L(s)  = 1  − 0.0857·2-s − 0.628·3-s − 0.992·4-s + 0.989·5-s + 0.0539·6-s − 0.261·7-s + 0.170·8-s − 0.604·9-s − 0.0849·10-s + 1.15·11-s + 0.624·12-s + 0.884·13-s + 0.0224·14-s − 0.622·15-s + 0.977·16-s + 0.617·17-s + 0.0518·18-s − 1.23·19-s − 0.982·20-s + 0.164·21-s − 0.0992·22-s + 0.208·23-s − 0.107·24-s − 0.0200·25-s − 0.0758·26-s + 1.00·27-s + 0.259·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(32.8195\)
Root analytic conductor: \(5.72883\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :15/2),\ -1)\)

Particular Values

\(L(8)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - 3.40e9T \)
good2 \( 1 + 15.5T + 3.27e4T^{2} \)
3 \( 1 + 2.38e3T + 1.43e7T^{2} \)
5 \( 1 - 1.72e5T + 3.05e10T^{2} \)
7 \( 1 + 5.70e5T + 4.74e12T^{2} \)
11 \( 1 - 7.47e7T + 4.17e15T^{2} \)
13 \( 1 - 2.00e8T + 5.11e16T^{2} \)
17 \( 1 - 1.04e9T + 2.86e18T^{2} \)
19 \( 1 + 4.82e9T + 1.51e19T^{2} \)
29 \( 1 + 9.04e10T + 8.62e21T^{2} \)
31 \( 1 + 1.68e11T + 2.34e22T^{2} \)
37 \( 1 + 5.36e11T + 3.33e23T^{2} \)
41 \( 1 + 1.56e12T + 1.55e24T^{2} \)
43 \( 1 - 2.10e12T + 3.17e24T^{2} \)
47 \( 1 + 2.98e12T + 1.20e25T^{2} \)
53 \( 1 - 5.68e11T + 7.31e25T^{2} \)
59 \( 1 + 2.01e13T + 3.65e26T^{2} \)
61 \( 1 - 2.55e13T + 6.02e26T^{2} \)
67 \( 1 + 3.39e13T + 2.46e27T^{2} \)
71 \( 1 - 1.82e13T + 5.87e27T^{2} \)
73 \( 1 - 1.58e14T + 8.90e27T^{2} \)
79 \( 1 - 1.90e14T + 2.91e28T^{2} \)
83 \( 1 + 2.14e14T + 6.11e28T^{2} \)
89 \( 1 + 7.76e14T + 1.74e29T^{2} \)
97 \( 1 + 7.97e14T + 6.33e29T^{2} \)
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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

−13.69909258141755209834389370540, −12.53207301194063727427075328837, −10.96900555712991335597684490239, −9.598954241682889721204441534059, −8.612093023289284161976168785111, −6.35984038656210863417579492511, −5.39431687505194592700925880933, −3.71172365605407017394431788874, −1.49232349245655031222281520048, 0, 1.49232349245655031222281520048, 3.71172365605407017394431788874, 5.39431687505194592700925880933, 6.35984038656210863417579492511, 8.612093023289284161976168785111, 9.598954241682889721204441534059, 10.96900555712991335597684490239, 12.53207301194063727427075328837, 13.69909258141755209834389370540

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