↑

Properties

Label 2-117-13.10-c1-0-2
Degree $2$
Conductor $117$
Sign $0.711 + 0.702i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.11i)2-s + (1.5 − 2.59i)4-s − 2.23i·5-s + (−3 − 1.73i)7-s + 2.23i·8-s + (2.50 + 4.33i)10-s + (3.87 − 2.23i)11-s + (2.5 − 2.59i)13-s + 7.74·14-s + (0.499 + 0.866i)16-s + (−1.93 + 3.35i)17-s + (−3 − 1.73i)19-s + (−5.80 − 3.35i)20-s + (−5 + 8.66i)22-s + (−3.87 − 6.70i)23-s + ⋯
L(s)  = 1  + (−1.36 + 0.790i)2-s + (0.750 − 1.29i)4-s − 0.999i·5-s + (−1.13 − 0.654i)7-s + 0.790i·8-s + (0.790 + 1.36i)10-s + (1.16 − 0.674i)11-s + (0.693 − 0.720i)13-s + 2.07·14-s + (0.124 + 0.216i)16-s + (−0.469 + 0.813i)17-s + (−0.688 − 0.397i)19-s + (−1.29 − 0.750i)20-s + (−1.06 + 1.84i)22-s + (−0.807 − 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431371 - 0.177061i\)
\(L(\frac12)\) \(\approx\) \(0.431371 - 0.177061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (1.93 - 1.11i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.23iT - 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.87 + 2.23i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.87 + 6.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.93 - 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.93 - 1.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-7.74 - 4.47i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.87 + 2.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + (-3.87 + 2.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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

−13.31037908861479624331183851989, −12.53689326781823293416925333684, −10.84730408745289793123264521631, −9.965883724133590426462790938592, −8.790544420883197459839928379689, −8.421944462953849146553962966170, −6.78302115143903205810651792722, −6.09029336514007183377741640321, −3.96776037969244900396210465355, −0.818688835262415973337208139605, 2.16870558560299373678782087364, 3.60509665832055310205515387398, 6.28404065868778545540930149869, 7.18895991060216929298625643263, 8.773008241680129230188723598600, 9.522708935914662603518602186021, 10.29343732212689778833423517934, 11.53644546317749212928972732484, 12.02811949727293546212105008242, 13.53675612417268250745462805170

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