↑

Properties

Label 2-117-117.103-c3-0-17
Degree $2$
Conductor $117$
Sign $0.794 - 0.606i$
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  + (−2.61 + 1.51i)2-s + (1.89 − 4.83i)3-s + (0.575 − 0.997i)4-s + (18.8 + 10.8i)5-s + (2.34 + 15.5i)6-s + (0.934 − 0.539i)7-s − 20.7i·8-s + (−19.7 − 18.3i)9-s − 65.7·10-s + (−25.6 + 14.8i)11-s + (−3.72 − 4.67i)12-s + (44.1 + 15.8i)13-s + (−1.63 + 2.82i)14-s + (88.2 − 70.3i)15-s + (35.9 + 62.2i)16-s + 85.5·17-s + ⋯
L(s)  = 1  + (−0.926 + 0.534i)2-s + (0.365 − 0.930i)3-s + (0.0719 − 0.124i)4-s + (1.68 + 0.971i)5-s + (0.159 + 1.05i)6-s + (0.0504 − 0.0291i)7-s − 0.915i·8-s + (−0.732 − 0.680i)9-s − 2.07·10-s + (−0.704 + 0.406i)11-s + (−0.0897 − 0.112i)12-s + (0.941 + 0.337i)13-s + (−0.0311 + 0.0539i)14-s + (1.51 − 1.21i)15-s + (0.561 + 0.972i)16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.794 - 0.606i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.32675 + 0.448691i\)
\(L(\frac12)\) \(\approx\) \(1.32675 + 0.448691i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.89 + 4.83i)T \)
13 \( 1 + (-44.1 - 15.8i)T \)
good2 \( 1 + (2.61 - 1.51i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-18.8 - 10.8i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-0.934 + 0.539i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (25.6 - 14.8i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 85.5T + 4.91e3T^{2} \)
19 \( 1 - 93.0iT - 6.85e3T^{2} \)
23 \( 1 + (-79.6 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (35.2 + 61.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-181. - 105. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 112. iT - 5.06e4T^{2} \)
41 \( 1 + (73.5 + 42.4i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (36.5 + 63.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (194. - 112. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 525.T + 1.48e5T^{2} \)
59 \( 1 + (148. + 85.9i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (22.9 + 39.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (778. + 449. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 569. iT - 3.57e5T^{2} \)
73 \( 1 - 148. iT - 3.89e5T^{2} \)
79 \( 1 + (434. + 752. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (440. - 254. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 448. iT - 7.04e5T^{2} \)
97 \( 1 + (245. - 142. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−13.32218411832180596391186446594, −12.40873315522427677294720537684, −10.60441366185125358445966362583, −9.841204121449020593660488447697, −8.770556770283315109717677243424, −7.68483312612980991084759930409, −6.65635131620356360399910572671, −5.88692538854361096667057204034, −3.03929206071338682818179165282, −1.49287620271769223547325590324, 1.19404931843646420632474739302, 2.78026766940216312284166052778, 5.09668346796101379699989274161, 5.68360142787738345623528202901, 8.251890074792872389448309211356, 8.956790254028236311107496581001, 9.792996188608425325292513043275, 10.36341843998096273243710691263, 11.46154095416265119400354034895, 13.29614429468867108685845543794

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