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  + 3.12·2-s − 5.79·3-s + 5.75·4-s − 18.0·6-s + 5.49·8-s + 24.5·9-s − 33.3·12-s − 2.15·13-s − 5.86·16-s + 76.7·18-s − 23·23-s − 31.8·24-s + 25·25-s − 6.72·26-s − 90.1·27-s − 13.0·29-s + 61.9·31-s − 40.3·32-s + 141.·36-s + 12.4·39-s − 66.2·41-s − 71.8·46-s + 50.9·47-s + 33.9·48-s + 49·49-s + 78.0·50-s − 12.4·52-s + ⋯
L(s)  = 1  + 1.56·2-s − 1.93·3-s + 1.43·4-s − 3.01·6-s + 0.686·8-s + 2.72·9-s − 2.78·12-s − 0.165·13-s − 0.366·16-s + 4.26·18-s − 23-s − 1.32·24-s + 25-s − 0.258·26-s − 3.33·27-s − 0.450·29-s + 1.99·31-s − 1.25·32-s + 3.92·36-s + 0.319·39-s − 1.61·41-s − 1.56·46-s + 1.08·47-s + 0.708·48-s + 0.999·49-s + 1.56·50-s − 0.238·52-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  =  $(2,\ 23,\ (\ :1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $1.15945$
$L(\frac12)$  $\approx$  $1.15945$
$L(2)$   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 \( 1 - 3.12T + 4T^{2} \)
3 \( 1 + 5.79T + 9T^{2} \)
5 \( 1 - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 2.15T + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361T^{2} \)
29 \( 1 + 13.0T + 841T^{2} \)
31 \( 1 - 61.9T + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 66.2T + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 50.9T + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 26T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 + 55.2T + 5.04e3T^{2} \)
73 \( 1 + 88.0T + 5.32e3T^{2} \)
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

−17.38662438077566725456787643412, −16.25266090259699302544844085089, −15.25045957856847525792594451566, −13.54348074276356119184964203974, −12.34454102311100891116481500791, −11.66311862169944965457286273302, −10.35740039286693641585425892371, −6.81039803586467718459107985552, −5.67469663404156687062790353124, −4.46007887250445186823029671800, 4.46007887250445186823029671800, 5.67469663404156687062790353124, 6.81039803586467718459107985552, 10.35740039286693641585425892371, 11.66311862169944965457286273302, 12.34454102311100891116481500791, 13.54348074276356119184964203974, 15.25045957856847525792594451566, 16.25266090259699302544844085089, 17.38662438077566725456787643412

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