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Properties

Label 2-23-23.22-c12-0-10
Degree $2$
Conductor $23$
Sign $0.900 + 0.434i$
Analytic cond. $21.0218$
Root an. cond. $4.58495$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.17·2-s − 251.·3-s − 4.01e3·4-s − 7.85e3i·5-s − 2.31e3·6-s + 1.86e5i·7-s − 7.43e4·8-s − 4.68e5·9-s − 7.20e4i·10-s − 1.55e6i·11-s + 1.01e6·12-s + 2.91e6·13-s + 1.71e6i·14-s + 1.97e6i·15-s + 1.57e7·16-s − 3.03e6i·17-s + ⋯
L(s)  = 1  + 0.143·2-s − 0.345·3-s − 0.979·4-s − 0.502i·5-s − 0.0495·6-s + 1.58i·7-s − 0.283·8-s − 0.880·9-s − 0.0720i·10-s − 0.880i·11-s + 0.338·12-s + 0.603·13-s + 0.227i·14-s + 0.173i·15-s + 0.938·16-s − 0.125i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(21.0218\)
Root analytic conductor: \(4.58495\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :6),\ 0.900 + 0.434i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.11666 - 0.255257i\)
\(L(\frac12)\) \(\approx\) \(1.11666 - 0.255257i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-1.33e8 - 6.43e7i)T \)
good2 \( 1 - 9.17T + 4.09e3T^{2} \)
3 \( 1 + 251.T + 5.31e5T^{2} \)
5 \( 1 + 7.85e3iT - 2.44e8T^{2} \)
7 \( 1 - 1.86e5iT - 1.38e10T^{2} \)
11 \( 1 + 1.55e6iT - 3.13e12T^{2} \)
13 \( 1 - 2.91e6T + 2.32e13T^{2} \)
17 \( 1 + 3.03e6iT - 5.82e14T^{2} \)
19 \( 1 + 4.66e7iT - 2.21e15T^{2} \)
29 \( 1 - 3.96e8T + 3.53e17T^{2} \)
31 \( 1 + 1.56e8T + 7.87e17T^{2} \)
37 \( 1 + 2.63e9iT - 6.58e18T^{2} \)
41 \( 1 - 1.40e9T + 2.25e19T^{2} \)
43 \( 1 + 2.75e9iT - 3.99e19T^{2} \)
47 \( 1 + 8.81e9T + 1.16e20T^{2} \)
53 \( 1 - 1.74e10iT - 4.91e20T^{2} \)
59 \( 1 - 2.10e10T + 1.77e21T^{2} \)
61 \( 1 - 8.13e10iT - 2.65e21T^{2} \)
67 \( 1 - 1.16e8iT - 8.18e21T^{2} \)
71 \( 1 - 7.66e10T + 1.64e22T^{2} \)
73 \( 1 - 1.64e11T + 2.29e22T^{2} \)
79 \( 1 + 3.14e11iT - 5.90e22T^{2} \)
83 \( 1 + 5.71e11iT - 1.06e23T^{2} \)
89 \( 1 - 7.59e11iT - 2.46e23T^{2} \)
97 \( 1 + 6.47e11iT - 6.93e23T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.85294822971995237328421331280, −13.53642888591041961037254646690, −12.38158312327078424488469703448, −11.19788647244676630898435745940, −9.013646063291327526139693992844, −8.633292080886513461347897156403, −5.90375371401909401899843064598, −5.04416105903473952223374746632, −2.99437512630028312356304558321, −0.62673635133560403747243697419, 0.871893014880532587682801726244, 3.48776654577360447855763401062, 4.83110463023590020372872947157, 6.62204225333037575564109403883, 8.211621881839685202677127093849, 9.938132795276230168696919063608, 10.98834635891875495582571474226, 12.70283446095169650532204389556, 13.92039455649197816296229313701, 14.69627379742479827179489092753

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