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  − 3.72·2-s + 4.24·3-s + 9.87·4-s − 15.8·6-s − 21.9·8-s + 9.05·9-s + 41.9·12-s − 21.3·13-s + 42.0·16-s − 33.7·18-s − 23·23-s − 93.0·24-s + 25·25-s + 79.5·26-s + 0.244·27-s + 55.4·29-s − 33.9·31-s − 69.1·32-s + 89.4·36-s − 90.7·39-s − 8.78·41-s + 85.6·46-s + 42.8·47-s + 178.·48-s + 49·49-s − 93.1·50-s − 211.·52-s + ⋯
L(s)  = 1  − 1.86·2-s + 1.41·3-s + 2.46·4-s − 2.63·6-s − 2.73·8-s + 1.00·9-s + 3.49·12-s − 1.64·13-s + 2.62·16-s − 1.87·18-s − 23-s − 3.87·24-s + 25-s + 3.06·26-s + 0.00906·27-s + 1.91·29-s − 1.09·31-s − 2.16·32-s + 2.48·36-s − 2.32·39-s − 0.214·41-s + 1.86·46-s + 0.912·47-s + 3.72·48-s + 0.999·49-s − 1.86·50-s − 4.05·52-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  =  $(2,\ 23,\ (\ :1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.613469$
$L(\frac12)$  $\approx$  $0.613469$
$L(2)$   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 \( 1 + 3.72T + 4T^{2} \)
3 \( 1 - 4.24T + 9T^{2} \)
5 \( 1 - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 21.3T + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361T^{2} \)
29 \( 1 - 55.4T + 841T^{2} \)
31 \( 1 + 33.9T + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 8.78T + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 42.8T + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 26T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 + 85.6T + 5.04e3T^{2} \)
73 \( 1 - 144.T + 5.32e3T^{2} \)
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

−17.86858580166154762691870631547, −16.66474855141397445291880866160, −15.38379949405769162888300515781, −14.30448934002337246511345373331, −12.18652506141296656813498136593, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −8.254627656874356514129115296000, −7.15878206375237697440477144189, −2.50316094689835549482408964468, 2.50316094689835549482408964468, 7.15878206375237697440477144189, 8.254627656874356514129115296000, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 12.18652506141296656813498136593, 14.30448934002337246511345373331, 15.38379949405769162888300515781, 16.66474855141397445291880866160, 17.86858580166154762691870631547

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