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Properties

Label 2-23-23.22-c24-0-46
Degree $2$
Conductor $23$
Sign $-0.987 - 0.159i$
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  + 6.81e3·2-s + 5.80e4·3-s + 2.96e7·4-s − 3.73e8i·5-s + 3.95e8·6-s + 1.51e10i·7-s + 8.78e10·8-s − 2.79e11·9-s − 2.54e12i·10-s + 3.53e11i·11-s + 1.72e12·12-s − 2.79e13·13-s + 1.03e14i·14-s − 2.16e13i·15-s + 1.00e14·16-s + 2.91e14i·17-s + ⋯
L(s)  = 1  + 1.66·2-s + 0.109·3-s + 1.76·4-s − 1.53i·5-s + 0.181·6-s + 1.09i·7-s + 1.27·8-s − 0.988·9-s − 2.54i·10-s + 0.112i·11-s + 0.193·12-s − 1.20·13-s + 1.82i·14-s − 0.167i·15-s + 0.358·16-s + 0.501i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.987 - 0.159i$
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: \((2,\ 23,\ (\ :12),\ -0.987 - 0.159i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.6818872055\)
\(L(\frac12)\) \(\approx\) \(0.6818872055\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (2.16e16 + 3.50e15i)T \)
good2 \( 1 - 6.81e3T + 1.67e7T^{2} \)
3 \( 1 - 5.80e4T + 2.82e11T^{2} \)
5 \( 1 + 3.73e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.51e10iT - 1.91e20T^{2} \)
11 \( 1 - 3.53e11iT - 9.84e24T^{2} \)
13 \( 1 + 2.79e13T + 5.42e26T^{2} \)
17 \( 1 - 2.91e14iT - 3.39e29T^{2} \)
19 \( 1 + 2.21e15iT - 4.89e30T^{2} \)
29 \( 1 - 1.53e17T + 1.25e35T^{2} \)
31 \( 1 + 7.94e16T + 6.20e35T^{2} \)
37 \( 1 - 4.84e18iT - 4.33e37T^{2} \)
41 \( 1 - 8.62e18T + 5.09e38T^{2} \)
43 \( 1 - 4.66e19iT - 1.59e39T^{2} \)
47 \( 1 + 2.01e20T + 1.35e40T^{2} \)
53 \( 1 - 2.41e19iT - 2.41e41T^{2} \)
59 \( 1 + 7.80e20T + 3.16e42T^{2} \)
61 \( 1 + 4.69e21iT - 7.04e42T^{2} \)
67 \( 1 - 7.26e20iT - 6.69e43T^{2} \)
71 \( 1 - 1.16e22T + 2.69e44T^{2} \)
73 \( 1 + 1.77e22T + 5.24e44T^{2} \)
79 \( 1 + 4.96e22iT - 3.49e45T^{2} \)
83 \( 1 + 6.77e22iT - 1.14e46T^{2} \)
89 \( 1 + 2.32e23iT - 6.10e46T^{2} \)
97 \( 1 - 8.76e23iT - 4.81e47T^{2} \)
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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

−12.29379946388962396176615476613, −11.58612298381138031164302161298, −9.337789381301841807343358344428, −8.200046546692857794899386576072, −6.21200432176774459973463398695, −5.20457739540954933353396234099, −4.58555147265390841176791878769, −2.95823431053193478513115310012, −1.95853109277450710473360827834, −0.07000123096819230833540171468, 2.26305826319127752636764871702, 3.14146956173507981566349534252, 4.05927162134843919910138158689, 5.55412528792294825966552234204, 6.68044986149546830984555726176, 7.59742123132488172341518074620, 10.11602600678783523761709769864, 11.15743693476742973707445953447, 12.15365615424243660779687002040, 13.79236258463290340204451200185

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