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Properties

Label 2-23-23.22-c12-0-21
Degree $2$
Conductor $23$
Sign $0.925 + 0.377i$
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  + 117.·2-s + 831.·3-s + 9.81e3·4-s − 2.62e4i·5-s + 9.80e4·6-s + 1.67e5i·7-s + 6.74e5·8-s + 1.59e5·9-s − 3.09e6i·10-s − 7.28e5i·11-s + 8.15e6·12-s − 5.83e6·13-s + 1.97e7i·14-s − 2.18e7i·15-s + 3.93e7·16-s − 1.39e7i·17-s + ⋯
L(s)  = 1  + 1.84·2-s + 1.14·3-s + 2.39·4-s − 1.68i·5-s + 2.10·6-s + 1.42i·7-s + 2.57·8-s + 0.300·9-s − 3.09i·10-s − 0.411i·11-s + 2.73·12-s − 1.20·13-s + 2.62i·14-s − 1.91i·15-s + 2.34·16-s − 0.576i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.925 + 0.377i$
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: \((2,\ 23,\ (\ :6),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(7.62567 - 1.49666i\)
\(L(\frac12)\) \(\approx\) \(7.62567 - 1.49666i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-1.37e8 - 5.59e7i)T \)
good2 \( 1 - 117.T + 4.09e3T^{2} \)
3 \( 1 - 831.T + 5.31e5T^{2} \)
5 \( 1 + 2.62e4iT - 2.44e8T^{2} \)
7 \( 1 - 1.67e5iT - 1.38e10T^{2} \)
11 \( 1 + 7.28e5iT - 3.13e12T^{2} \)
13 \( 1 + 5.83e6T + 2.32e13T^{2} \)
17 \( 1 + 1.39e7iT - 5.82e14T^{2} \)
19 \( 1 - 8.30e7iT - 2.21e15T^{2} \)
29 \( 1 - 6.50e8T + 3.53e17T^{2} \)
31 \( 1 - 4.47e7T + 7.87e17T^{2} \)
37 \( 1 + 1.55e8iT - 6.58e18T^{2} \)
41 \( 1 + 9.18e7T + 2.25e19T^{2} \)
43 \( 1 + 1.58e9iT - 3.99e19T^{2} \)
47 \( 1 + 4.29e9T + 1.16e20T^{2} \)
53 \( 1 - 1.15e10iT - 4.91e20T^{2} \)
59 \( 1 + 4.07e10T + 1.77e21T^{2} \)
61 \( 1 + 2.44e10iT - 2.65e21T^{2} \)
67 \( 1 + 1.58e11iT - 8.18e21T^{2} \)
71 \( 1 + 9.47e10T + 1.64e22T^{2} \)
73 \( 1 - 9.19e10T + 2.29e22T^{2} \)
79 \( 1 + 3.66e11iT - 5.90e22T^{2} \)
83 \( 1 - 1.44e11iT - 1.06e23T^{2} \)
89 \( 1 - 3.73e11iT - 2.46e23T^{2} \)
97 \( 1 - 9.37e11iT - 6.93e23T^{2} \)
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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

−14.68544975234311762165071072343, −13.66651280276226030259886642356, −12.47595952781660925230652726918, −11.99101710711647578835548391607, −9.264230338655390577651329813138, −8.020019138221395527205487952925, −5.71077650881709188066926053011, −4.77445851576055280282702967012, −3.12897093867486722238813388483, −1.91907572538567242690637150737, 2.46253713622626574040775438022, 3.18463434874542264332409230549, 4.48286086969159949069865256573, 6.77706073761335190060683965368, 7.36608661242789599102369102154, 10.25713462807484813441425985025, 11.31211864133731140631280053125, 13.11639380200172524221036547365, 14.04623184348592251940878215391, 14.64556189918412434430578051732

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