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Properties

Label 2-117-117.43-c3-0-26
Degree $2$
Conductor $117$
Sign $0.491 + 0.870i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.17 + 1.83i)2-s + (4.08 + 3.21i)3-s + (2.73 − 4.74i)4-s + (−15.6 + 9.05i)5-s + (−18.8 − 2.72i)6-s − 31.1i·7-s − 9.27i·8-s + (6.32 + 26.2i)9-s + (33.2 − 57.5i)10-s + (4.59 − 2.65i)11-s + (26.4 − 10.5i)12-s + (−38.4 − 26.7i)13-s + (57.2 + 99.1i)14-s + (−93.1 − 13.4i)15-s + (38.9 + 67.3i)16-s + (−1.95 − 3.38i)17-s + ⋯
L(s)  = 1  + (−1.12 + 0.648i)2-s + (0.785 + 0.618i)3-s + (0.342 − 0.592i)4-s + (−1.40 + 0.809i)5-s + (−1.28 − 0.185i)6-s − 1.68i·7-s − 0.409i·8-s + (0.234 + 0.972i)9-s + (1.05 − 1.82i)10-s + (0.125 − 0.0726i)11-s + (0.635 − 0.253i)12-s + (−0.821 − 0.570i)13-s + (1.09 + 1.89i)14-s + (−1.60 − 0.231i)15-s + (0.608 + 1.05i)16-s + (−0.0279 − 0.0483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.491 + 0.870i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.491 + 0.870i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.294953 - 0.172140i\)
\(L(\frac12)\) \(\approx\) \(0.294953 - 0.172140i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.08 - 3.21i)T \)
13 \( 1 + (38.4 + 26.7i)T \)
good2 \( 1 + (3.17 - 1.83i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (15.6 - 9.05i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + 31.1iT - 343T^{2} \)
11 \( 1 + (-4.59 + 2.65i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.5 + 11.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 55.9T + 1.21e4T^{2} \)
29 \( 1 + (147. + 255. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-150. + 86.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (328. + 189. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 132. iT - 6.89e4T^{2} \)
43 \( 1 + 60.6T + 7.95e4T^{2} \)
47 \( 1 + (229. + 132. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 446.T + 1.48e5T^{2} \)
59 \( 1 + (-219. - 126. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 - 352.T + 2.26e5T^{2} \)
67 \( 1 + 447. iT - 3.00e5T^{2} \)
71 \( 1 + (260. - 150. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 86.9iT - 3.89e5T^{2} \)
79 \( 1 + (243. - 422. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (842. + 486. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (789. + 456. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 13.7iT - 9.12e5T^{2} \)
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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

−13.10625510867972711465259072015, −11.34898249134423648769037237822, −10.40559152026389537158149542494, −9.719861350947264091097097751142, −8.231597161812514722684852634535, −7.57669456517381319593068527301, −7.00874412836072830923054724053, −4.30095459797287755424716906710, −3.39048457692158079982437674225, −0.24458233470369269732277810847, 1.61515349331188135410319706490, 3.08125497280436309953469420912, 5.06466868257202010101935916741, 7.17386933289655535543409827758, 8.386551474078985932511958955996, 8.748429016541772475752591343443, 9.614295013799001118270314106064, 11.42032428400600935027223256721, 12.11447920173760445691926805758, 12.61741434731150584271444867596

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