↑

Properties

Label 2-117-13.9-c3-0-7
Degree $2$
Conductor $117$
Sign $0.966 - 0.255i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.28 + 3.95i)2-s + (−6.40 − 11.0i)4-s − 2.80·5-s + (−4.78 − 8.28i)7-s + 21.9·8-s + (6.40 − 11.0i)10-s + (19.7 − 34.1i)11-s + (40.5 − 23.5i)13-s + 43.6·14-s + (1.21 − 2.09i)16-s + (1.00 + 1.74i)17-s + (30.0 + 52.1i)19-s + (17.9 + 31.1i)20-s + (89.9 + 155. i)22-s + (2.23 − 3.87i)23-s + ⋯
L(s)  = 1  + (−0.806 + 1.39i)2-s + (−0.800 − 1.38i)4-s − 0.251·5-s + (−0.258 − 0.447i)7-s + 0.969·8-s + (0.202 − 0.350i)10-s + (0.540 − 0.935i)11-s + (0.864 − 0.502i)13-s + 0.832·14-s + (0.0189 − 0.0327i)16-s + (0.0143 + 0.0248i)17-s + (0.363 + 0.629i)19-s + (0.201 + 0.348i)20-s + (0.871 + 1.50i)22-s + (0.0202 − 0.0350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.966 - 0.255i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.966 - 0.255i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.814604 + 0.105743i\)
\(L(\frac12)\) \(\approx\) \(0.814604 + 0.105743i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-40.5 + 23.5i)T \)
good2 \( 1 + (2.28 - 3.95i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 2.80T + 125T^{2} \)
7 \( 1 + (4.78 + 8.28i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-19.7 + 34.1i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-1.00 - 1.74i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-30.0 - 52.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-2.23 + 3.87i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.3 + 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 136.T + 2.97e4T^{2} \)
37 \( 1 + (-92.8 + 160. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-155. + 268. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (213. + 370. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 + 612.T + 1.48e5T^{2} \)
59 \( 1 + (258. + 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 - 139. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-24.9 + 43.2i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-139. - 242. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 467.T + 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1T + 5.71e5T^{2} \)
89 \( 1 + (-101. + 175. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-587. - 1.01e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−13.57113712160110848401270387524, −11.99881716856588620138698979167, −10.69942068614372338393421520937, −9.587323920624907293278998328535, −8.498290855953406932059923049203, −7.73845250006464880020502962043, −6.49964946354975196595654909878, −5.65491551289112755107606978597, −3.72347384732275396379261576408, −0.64953837019483212290604727153, 1.39681208255563442820084964064, 2.93772806626954416843494108998, 4.35969519495631296521279912842, 6.41561199469595640826864858314, 8.001110152086606539553685311595, 9.149672471637807427166131400978, 9.775893215389128979278685011546, 11.02509755650489482213384277138, 11.78783925764213768551444097734, 12.55972882097457973688208329734

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