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  − 1.61·2-s + 2.23·3-s + 0.618·4-s − 3.23·5-s − 3.61·6-s − 1.23·7-s + 2.23·8-s + 2.00·9-s + 5.23·10-s − 0.763·11-s + 1.38·12-s + 3·13-s + 2.00·14-s − 7.23·15-s − 4.85·16-s + 5.23·17-s − 3.23·18-s − 2·19-s − 2.00·20-s − 2.76·21-s + 1.23·22-s + 23-s + 5.00·24-s + 5.47·25-s − 4.85·26-s − 2.23·27-s − 0.763·28-s + ⋯
L(s)  = 1  − 1.14·2-s + 1.29·3-s + 0.309·4-s − 1.44·5-s − 1.47·6-s − 0.467·7-s + 0.790·8-s + 0.666·9-s + 1.65·10-s − 0.230·11-s + 0.398·12-s + 0.832·13-s + 0.534·14-s − 1.86·15-s − 1.21·16-s + 1.26·17-s − 0.762·18-s − 0.458·19-s − 0.447·20-s − 0.603·21-s + 0.263·22-s + 0.208·23-s + 1.02·24-s + 1.09·25-s − 0.951·26-s − 0.430·27-s − 0.144·28-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  =  $(2,\ 23,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.450379$
$L(\frac12)$  $\approx$  $0.450379$
$L(\frac{3}{2})$   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 \( 1 + 1.61T + 2T^{2} \)
3 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 + 0.763T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 2.47T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 - 7.76T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 - 6.94T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 - 4.29T + 97T^{2} \)
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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

−18.47750850800061646009902647786, −16.63291171064781582535205855744, −15.65538270528692681049757116794, −14.40066184005226474772481053609, −12.90390695537994036129091050107, −11.06445498538988679371642241803, −9.462792302698072156583043638081, −8.321948106011545948491366974293, −7.56305052056930034956656701516, −3.67521748643984848866430372052, 3.67521748643984848866430372052, 7.56305052056930034956656701516, 8.321948106011545948491366974293, 9.462792302698072156583043638081, 11.06445498538988679371642241803, 12.90390695537994036129091050107, 14.40066184005226474772481053609, 15.65538270528692681049757116794, 16.63291171064781582535205855744, 18.47750850800061646009902647786

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