Properties

Degree 2
Conductor 23
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 2.23·3-s − 1.61·4-s + 1.23·5-s − 1.38·6-s + 3.23·7-s − 2.23·8-s + 2.00·9-s + 0.763·10-s − 5.23·11-s + 3.61·12-s + 3·13-s + 2.00·14-s − 2.76·15-s + 1.85·16-s + 0.763·17-s + 1.23·18-s − 2·19-s − 2.00·20-s − 7.23·21-s − 3.23·22-s + 23-s + 5.00·24-s − 3.47·25-s + 1.85·26-s + 2.23·27-s − 5.23·28-s + ⋯
L(s)  = 1  + 0.437·2-s − 1.29·3-s − 0.809·4-s + 0.552·5-s − 0.564·6-s + 1.22·7-s − 0.790·8-s + 0.666·9-s + 0.241·10-s − 1.57·11-s + 1.04·12-s + 0.832·13-s + 0.534·14-s − 0.713·15-s + 0.463·16-s + 0.185·17-s + 0.291·18-s − 0.458·19-s − 0.447·20-s − 1.57·21-s − 0.689·22-s + 0.208·23-s + 1.02·24-s − 0.694·25-s + 0.363·26-s + 0.430·27-s − 0.989·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{23} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 23,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.551605$
$L(\frac12)$  $\approx$  $0.551605$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 23$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 23$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad23 \( 1 - T \)
good2 \( 1 - 0.618T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 - 3.23T + 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 + 1.23T + 37T^{2} \)
41 \( 1 + 3.47T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 + 6.94T + 61T^{2} \)
67 \( 1 + 2.76T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 6.52T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 8.76T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.90710782011974663086496588020, −17.08205862019940285982025794172, −15.46280104465881945233209269948, −13.96241428320228018659273252765, −12.89016809230386862701948159604, −11.46302391247974405152532879935, −10.29026506954357994195204927063, −8.278190574714781685082992842254, −5.81727025604144459401902615206, −4.87645242907439515159200857913, 4.87645242907439515159200857913, 5.81727025604144459401902615206, 8.278190574714781685082992842254, 10.29026506954357994195204927063, 11.46302391247974405152532879935, 12.89016809230386862701948159604, 13.96241428320228018659273252765, 15.46280104465881945233209269948, 17.08205862019940285982025794172, 17.90710782011974663086496588020

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