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Properties

Label 2-23-23.22-c24-0-3
Degree $2$
Conductor $23$
Sign $-0.888 + 0.457i$
Analytic cond. $83.9424$
Root an. cond. $9.16201$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.58e3·2-s + 7.76e5·3-s + 4.07e7·4-s + 3.88e8i·5-s − 5.88e9·6-s − 7.00e9i·7-s − 1.81e11·8-s + 3.20e11·9-s − 2.94e12i·10-s + 3.53e12i·11-s + 3.16e13·12-s + 7.20e12·13-s + 5.31e13i·14-s + 3.01e14i·15-s + 6.95e14·16-s − 3.92e13i·17-s + ⋯
L(s)  = 1  − 1.85·2-s + 1.46·3-s + 2.42·4-s + 1.59i·5-s − 2.70·6-s − 0.506i·7-s − 2.64·8-s + 1.13·9-s − 2.94i·10-s + 1.12i·11-s + 3.54·12-s + 0.309·13-s + 0.937i·14-s + 2.32i·15-s + 2.47·16-s − 0.0673i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.457i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.888 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.888 + 0.457i$
Analytic conductor: \(83.9424\)
Root analytic conductor: \(9.16201\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :12),\ -0.888 + 0.457i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.3479313054\)
\(L(\frac12)\) \(\approx\) \(0.3479313054\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.94e16 - 1.00e16i)T \)
good2 \( 1 + 7.58e3T + 1.67e7T^{2} \)
3 \( 1 - 7.76e5T + 2.82e11T^{2} \)
5 \( 1 - 3.88e8iT - 5.96e16T^{2} \)
7 \( 1 + 7.00e9iT - 1.91e20T^{2} \)
11 \( 1 - 3.53e12iT - 9.84e24T^{2} \)
13 \( 1 - 7.20e12T + 5.42e26T^{2} \)
17 \( 1 + 3.92e13iT - 3.39e29T^{2} \)
19 \( 1 + 2.86e15iT - 4.89e30T^{2} \)
29 \( 1 + 1.47e17T + 1.25e35T^{2} \)
31 \( 1 + 4.74e17T + 6.20e35T^{2} \)
37 \( 1 - 1.14e19iT - 4.33e37T^{2} \)
41 \( 1 + 3.61e19T + 5.09e38T^{2} \)
43 \( 1 + 7.05e19iT - 1.59e39T^{2} \)
47 \( 1 + 1.72e20T + 1.35e40T^{2} \)
53 \( 1 - 5.62e20iT - 2.41e41T^{2} \)
59 \( 1 - 3.17e21T + 3.16e42T^{2} \)
61 \( 1 + 4.66e21iT - 7.04e42T^{2} \)
67 \( 1 - 3.10e21iT - 6.69e43T^{2} \)
71 \( 1 + 1.62e22T + 2.69e44T^{2} \)
73 \( 1 + 2.51e21T + 5.24e44T^{2} \)
79 \( 1 - 7.84e22iT - 3.49e45T^{2} \)
83 \( 1 - 3.55e22iT - 1.14e46T^{2} \)
89 \( 1 - 1.67e22iT - 6.10e46T^{2} \)
97 \( 1 + 2.72e23iT - 4.81e47T^{2} \)
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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

−13.69992884456746132411197568667, −11.47851292821426045520039564043, −10.30000125910279344437321608115, −9.637989731795930314121846314997, −8.391026660715717673435569873126, −7.31042262084960011747133357620, −6.82044644731775482137985670529, −3.50179841332748100152375863255, −2.49695949771008637490252763781, −1.71558762380324353214966943465, 0.11133503992824602544037556962, 1.31333958356714595462460701017, 2.12071060575234181922785269805, 3.58344147058115241054300265895, 5.85677481640231471054489370410, 7.82617748554845637654079158422, 8.519892006305117817001377068472, 8.948415617181500482320287974686, 10.00906427278279110549354407021, 11.71246620216612725152812069223

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