Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.601·2-s + 1.54·3-s − 3.63·4-s + 0.928·6-s − 4.59·8-s − 6.61·9-s − 5.61·12-s + 23.5·13-s + 11.7·16-s − 3.98·18-s − 23·23-s − 7.09·24-s + 25·25-s + 14.1·26-s − 24.1·27-s − 42.4·29-s − 27.9·31-s + 25.4·32-s + 24.0·36-s + 36.3·39-s + 74.9·41-s − 13.8·46-s − 93.8·47-s + 18.1·48-s + 49·49-s + 15.0·50-s − 85.5·52-s + ⋯
L(s)  = 1  + 0.300·2-s + 0.514·3-s − 0.909·4-s + 0.154·6-s − 0.574·8-s − 0.735·9-s − 0.468·12-s + 1.80·13-s + 0.736·16-s − 0.221·18-s − 23-s − 0.295·24-s + 25-s + 0.544·26-s − 0.892·27-s − 1.46·29-s − 0.902·31-s + 0.795·32-s + 0.668·36-s + 0.930·39-s + 1.82·41-s − 0.300·46-s − 1.99·47-s + 0.379·48-s + 0.999·49-s + 0.300·50-s − 1.64·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{23} (22, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 23,\ (\ :1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.989480$
$L(\frac12)$  $\approx$  $0.989480$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 23$, \(F_p\) is a polynomial of degree 2. If $p = 23$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad23 \( 1 + 23T \)
good2 \( 1 - 0.601T + 4T^{2} \)
3 \( 1 - 1.54T + 9T^{2} \)
5 \( 1 - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 - 23.5T + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361T^{2} \)
29 \( 1 + 42.4T + 841T^{2} \)
31 \( 1 + 27.9T + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 74.9T + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + 93.8T + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 26T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 140.T + 5.04e3T^{2} \)
73 \( 1 + 56.8T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−17.87908189478010385509079805331, −16.38854251263791914136378027948, −14.82823558841975961978977417582, −13.87709638210915675759205960405, −12.88326648893801043752994094227, −11.12595221769137640392299636957, −9.255683549229959107838401341587, −8.249320660797426125572295542019, −5.79533420479259215020430426704, −3.69917499700457382202237665871, 3.69917499700457382202237665871, 5.79533420479259215020430426704, 8.249320660797426125572295542019, 9.255683549229959107838401341587, 11.12595221769137640392299636957, 12.88326648893801043752994094227, 13.87709638210915675759205960405, 14.82823558841975961978977417582, 16.38854251263791914136378027948, 17.87908189478010385509079805331

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