Properties

Label 2-1232-308.263-c1-0-39
Degree $2$
Conductor $1232$
Sign $0.352 + 0.935i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 0.784i)3-s + (−1.10 − 1.92i)5-s + (2.33 − 1.24i)7-s + (−0.269 − 0.466i)9-s + (3.30 + 0.220i)11-s − 0.689i·13-s − 3.48i·15-s + (−4.42 − 2.55i)17-s + (−3.83 − 6.64i)19-s + (4.14 + 0.137i)21-s + (−4.29 + 2.48i)23-s + (0.0360 − 0.0624i)25-s − 5.55i·27-s + 2.89i·29-s + (5.75 + 3.32i)31-s + ⋯
L(s)  = 1  + (0.784 + 0.452i)3-s + (−0.496 − 0.859i)5-s + (0.882 − 0.471i)7-s + (−0.0897 − 0.155i)9-s + (0.997 + 0.0664i)11-s − 0.191i·13-s − 0.899i·15-s + (−1.07 − 0.619i)17-s + (−0.880 − 1.52i)19-s + (0.905 + 0.0299i)21-s + (−0.896 + 0.517i)23-s + (0.00720 − 0.0124i)25-s − 1.06i·27-s + 0.536i·29-s + (1.03 + 0.596i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965122702\)
\(L(\frac12)\) \(\approx\) \(1.965122702\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.33 + 1.24i)T \)
11 \( 1 + (-3.30 - 0.220i)T \)
good3 \( 1 + (-1.35 - 0.784i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.10 + 1.92i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.689iT - 13T^{2} \)
17 \( 1 + (4.42 + 2.55i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.83 + 6.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.89iT - 29T^{2} \)
31 \( 1 + (-5.75 - 3.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.36 - 7.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.78iT - 41T^{2} \)
43 \( 1 + 6.15T + 43T^{2} \)
47 \( 1 + (-4.05 + 2.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.52 - 2.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.13 + 0.658i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.68 + 1.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.14 - 0.663i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.14iT - 71T^{2} \)
73 \( 1 + (-12.3 - 7.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.66 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.98T + 83T^{2} \)
89 \( 1 + (-2.10 - 3.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−9.304762600289813623511800866816, −8.641814213457991126478441149273, −8.338318201342176747863365974553, −7.18532876582837014965198996741, −6.35785808769824393307056224514, −4.78677307111276579974428127689, −4.48946662889106651754092984360, −3.53829667561545094264189384447, −2.24001979902149813301107156590, −0.76816755093466679058251823189, 1.77283455374265874494456994579, 2.46012489028587054400624454245, 3.74692524791492089801398470031, 4.45143064383885995542633442823, 5.95773728338251271870388195206, 6.58328814059757799765303756359, 7.65847206249925677123499489828, 8.219571438560374968906539905288, 8.739978760993893331142251963213, 9.817525872312074308685799342743

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