↑

Properties

Label 2-117-13.8-c4-0-20
Degree $2$
Conductor $117$
Sign $0.738 + 0.673i$
Analytic cond. $12.0942$
Root an. cond. $3.47768$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.05 + 4.05i)2-s + 16.8i·4-s + (−33.5 − 33.5i)5-s + (23.7 − 23.7i)7-s + (−3.29 + 3.29i)8-s − 271. i·10-s + (−32.2 + 32.2i)11-s + (72.8 − 152. i)13-s + 192.·14-s + 242.·16-s − 354. i·17-s + (−168. − 168. i)19-s + (564. − 564. i)20-s − 261.·22-s + 182. i·23-s + ⋯
L(s)  = 1  + (1.01 + 1.01i)2-s + 1.05i·4-s + (−1.34 − 1.34i)5-s + (0.483 − 0.483i)7-s + (−0.0514 + 0.0514i)8-s − 2.71i·10-s + (−0.266 + 0.266i)11-s + (0.430 − 0.902i)13-s + 0.980·14-s + 0.946·16-s − 1.22i·17-s + (−0.466 − 0.466i)19-s + (1.41 − 1.41i)20-s − 0.539·22-s + 0.344i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.738 + 0.673i$
Analytic conductor: \(12.0942\)
Root analytic conductor: \(3.47768\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :2),\ 0.738 + 0.673i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.97847 - 0.766636i\)
\(L(\frac12)\) \(\approx\) \(1.97847 - 0.766636i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-72.8 + 152. i)T \)
good2 \( 1 + (-4.05 - 4.05i)T + 16iT^{2} \)
5 \( 1 + (33.5 + 33.5i)T + 625iT^{2} \)
7 \( 1 + (-23.7 + 23.7i)T - 2.40e3iT^{2} \)
11 \( 1 + (32.2 - 32.2i)T - 1.46e4iT^{2} \)
17 \( 1 + 354. iT - 8.35e4T^{2} \)
19 \( 1 + (168. + 168. i)T + 1.30e5iT^{2} \)
23 \( 1 - 182. iT - 2.79e5T^{2} \)
29 \( 1 + 913.T + 7.07e5T^{2} \)
31 \( 1 + (876. + 876. i)T + 9.23e5iT^{2} \)
37 \( 1 + (-112. + 112. i)T - 1.87e6iT^{2} \)
41 \( 1 + (-1.13e3 - 1.13e3i)T + 2.82e6iT^{2} \)
43 \( 1 - 411. iT - 3.41e6T^{2} \)
47 \( 1 + (-3.00e3 + 3.00e3i)T - 4.87e6iT^{2} \)
53 \( 1 - 4.22e3T + 7.89e6T^{2} \)
59 \( 1 + (1.69e3 - 1.69e3i)T - 1.21e7iT^{2} \)
61 \( 1 + 3.76e3T + 1.38e7T^{2} \)
67 \( 1 + (2.63e3 + 2.63e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (297. + 297. i)T + 2.54e7iT^{2} \)
73 \( 1 + (-2.61e3 + 2.61e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 7.88e3T + 3.89e7T^{2} \)
83 \( 1 + (-4.07e3 - 4.07e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (1.72e3 - 1.72e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (-1.56e3 - 1.56e3i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.97719587575775164965098459438, −12.03123548897729228585721201284, −10.92600651679562496918053670880, −9.112881458096535919528232882342, −7.81387200838757261499241842470, −7.40049722584564061025484892190, −5.53784368893425124147865912966, −4.68027116455862546819842741642, −3.77620605815195517809073513804, −0.68255469668027603357802883204, 2.11140452653785743868614569657, 3.48558578847526040258010799633, 4.24336474678172585420890717020, 5.95992193999555424446988773991, 7.42173163908801415453349379149, 8.545483625221009716861908137122, 10.62350213088165557350400677640, 10.98431357391549642770172923537, 11.88365618620533870885300170488, 12.63804093303472826257341656296

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