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Properties

Label 2-23-23.22-c12-0-1
Degree $2$
Conductor $23$
Sign $0.193 - 0.981i$
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  + 52.2·2-s − 1.13e3·3-s − 1.36e3·4-s − 2.60e4i·5-s − 5.95e4·6-s − 1.51e4i·7-s − 2.85e5·8-s + 7.65e5·9-s − 1.36e6i·10-s + 5.55e4i·11-s + 1.55e6·12-s − 2.41e6·13-s − 7.92e5i·14-s + 2.97e7i·15-s − 9.33e6·16-s + 1.88e7i·17-s + ⋯
L(s)  = 1  + 0.816·2-s − 1.56·3-s − 0.332·4-s − 1.67i·5-s − 1.27·6-s − 0.128i·7-s − 1.08·8-s + 1.44·9-s − 1.36i·10-s + 0.0313i·11-s + 0.520·12-s − 0.500·13-s − 0.105i·14-s + 2.60i·15-s − 0.556·16-s + 0.781i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.193 - 0.981i$
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: \((2,\ 23,\ (\ :6),\ 0.193 - 0.981i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.299145 + 0.245832i\)
\(L(\frac12)\) \(\approx\) \(0.299145 + 0.245832i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-2.86e7 + 1.45e8i)T \)
good2 \( 1 - 52.2T + 4.09e3T^{2} \)
3 \( 1 + 1.13e3T + 5.31e5T^{2} \)
5 \( 1 + 2.60e4iT - 2.44e8T^{2} \)
7 \( 1 + 1.51e4iT - 1.38e10T^{2} \)
11 \( 1 - 5.55e4iT - 3.13e12T^{2} \)
13 \( 1 + 2.41e6T + 2.32e13T^{2} \)
17 \( 1 - 1.88e7iT - 5.82e14T^{2} \)
19 \( 1 - 4.59e7iT - 2.21e15T^{2} \)
29 \( 1 + 1.19e6T + 3.53e17T^{2} \)
31 \( 1 - 1.39e9T + 7.87e17T^{2} \)
37 \( 1 - 3.52e9iT - 6.58e18T^{2} \)
41 \( 1 + 4.39e9T + 2.25e19T^{2} \)
43 \( 1 + 1.17e10iT - 3.99e19T^{2} \)
47 \( 1 + 6.12e9T + 1.16e20T^{2} \)
53 \( 1 - 3.46e10iT - 4.91e20T^{2} \)
59 \( 1 + 5.57e10T + 1.77e21T^{2} \)
61 \( 1 - 2.03e10iT - 2.65e21T^{2} \)
67 \( 1 + 6.67e10iT - 8.18e21T^{2} \)
71 \( 1 - 3.36e8T + 1.64e22T^{2} \)
73 \( 1 + 1.87e11T + 2.29e22T^{2} \)
79 \( 1 - 4.42e11iT - 5.90e22T^{2} \)
83 \( 1 + 7.73e10iT - 1.06e23T^{2} \)
89 \( 1 - 4.81e11iT - 2.46e23T^{2} \)
97 \( 1 + 5.07e11iT - 6.93e23T^{2} \)
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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

−15.42501954437200219602954963375, −13.59903407582421950528590525253, −12.36716605112150582072550060342, −12.10367308089256432438681136825, −10.14906303674363133850888836256, −8.528269419479754001816672051635, −6.14257529870526170367094542256, −5.09590052714998944417264543257, −4.31600113911864735025288425464, −1.02808635683658465834380338158, 0.16331156915664030566296883146, 2.97831215205130235497133477521, 4.76101836480775353914792046884, 6.00746956971596104884258996187, 7.04954414504806277966814928390, 9.816850581394834147270988215974, 11.15861953904453185165274318826, 11.92102022694097439899828170365, 13.43704595334801940616349677330, 14.67138680647120217643656150895

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