Properties

Label 2-23-1.1-c17-0-8
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $42.1410$
Root an. cond. $6.49161$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 174.·2-s − 1.54e4·3-s − 1.00e5·4-s − 1.53e6·5-s + 2.68e6·6-s − 1.10e7·7-s + 4.03e7·8-s + 1.09e8·9-s + 2.67e8·10-s + 7.06e8·11-s + 1.55e9·12-s − 1.64e9·13-s + 1.91e9·14-s + 2.37e10·15-s + 6.16e9·16-s − 9.14e9·17-s − 1.90e10·18-s − 3.02e10·19-s + 1.54e11·20-s + 1.69e11·21-s − 1.23e11·22-s − 7.83e10·23-s − 6.23e11·24-s + 1.60e12·25-s + 2.85e11·26-s + 3.08e11·27-s + 1.10e12·28-s + ⋯
L(s)  = 1  − 0.481·2-s − 1.35·3-s − 0.768·4-s − 1.76·5-s + 0.653·6-s − 0.721·7-s + 0.851·8-s + 0.845·9-s + 0.847·10-s + 0.994·11-s + 1.04·12-s − 0.557·13-s + 0.347·14-s + 2.39·15-s + 0.358·16-s − 0.317·17-s − 0.406·18-s − 0.408·19-s + 1.35·20-s + 0.979·21-s − 0.478·22-s − 0.208·23-s − 1.15·24-s + 2.10·25-s + 0.268·26-s + 0.210·27-s + 0.554·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(42.1410\)
Root analytic conductor: \(6.49161\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :17/2),\ -1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + 7.83e10T \)
good2 \( 1 + 174.T + 1.31e5T^{2} \)
3 \( 1 + 1.54e4T + 1.29e8T^{2} \)
5 \( 1 + 1.53e6T + 7.62e11T^{2} \)
7 \( 1 + 1.10e7T + 2.32e14T^{2} \)
11 \( 1 - 7.06e8T + 5.05e17T^{2} \)
13 \( 1 + 1.64e9T + 8.65e18T^{2} \)
17 \( 1 + 9.14e9T + 8.27e20T^{2} \)
19 \( 1 + 3.02e10T + 5.48e21T^{2} \)
29 \( 1 + 1.73e12T + 7.25e24T^{2} \)
31 \( 1 - 4.09e11T + 2.25e25T^{2} \)
37 \( 1 - 2.45e13T + 4.56e26T^{2} \)
41 \( 1 - 4.13e13T + 2.61e27T^{2} \)
43 \( 1 - 4.40e13T + 5.87e27T^{2} \)
47 \( 1 + 3.15e14T + 2.66e28T^{2} \)
53 \( 1 - 3.38e14T + 2.05e29T^{2} \)
59 \( 1 + 1.65e15T + 1.27e30T^{2} \)
61 \( 1 - 1.88e15T + 2.24e30T^{2} \)
67 \( 1 + 3.94e15T + 1.10e31T^{2} \)
71 \( 1 - 9.31e15T + 2.96e31T^{2} \)
73 \( 1 - 1.50e15T + 4.74e31T^{2} \)
79 \( 1 - 2.52e16T + 1.81e32T^{2} \)
83 \( 1 - 2.69e16T + 4.21e32T^{2} \)
89 \( 1 - 7.01e16T + 1.37e33T^{2} \)
97 \( 1 + 6.70e16T + 5.95e33T^{2} \)
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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

−12.79628558715962558783180576948, −11.85412317379327265898182405950, −10.86594465643882765570182246115, −9.347780324244358978459139862806, −7.86564039859984670580087028581, −6.53525704898095400395476414564, −4.75960401657025647632905403876, −3.78049635718686893592436336604, −0.75607542427628276910396769241, 0, 0.75607542427628276910396769241, 3.78049635718686893592436336604, 4.75960401657025647632905403876, 6.53525704898095400395476414564, 7.86564039859984670580087028581, 9.347780324244358978459139862806, 10.86594465643882765570182246115, 11.85412317379327265898182405950, 12.79628558715962558783180576948

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