Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{23} (22, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 23,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.1740363269$
$L(\frac12)$  $\approx$  $0.1740363269$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 23$, \(F_p\) is a polynomial of degree 2. If $p = 23$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad23 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−18.08066174344421407705280597229, −17.10126330358183155194872344463, −16.49823251336240134157531665336, −14.66248095457465912598048536284, −12.93440966771841183579118145714, −11.43003635304813311896205990224, −10.28202740085213205371618003800, −8.881396573689177474965824491608, −7.15926229054170384129282782025, −5.11568332881511759855335642038, 5.11568332881511759855335642038, 7.15926229054170384129282782025, 8.881396573689177474965824491608, 10.28202740085213205371618003800, 11.43003635304813311896205990224, 12.93440966771841183579118145714, 14.66248095457465912598048536284, 16.49823251336240134157531665336, 17.10126330358183155194872344463, 18.08066174344421407705280597229

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