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Properties

Label 2-117-117.103-c3-0-37
Degree $2$
Conductor $117$
Sign $0.154 + 0.987i$
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  + (4.04 − 2.33i)2-s + (5.18 − 0.308i)3-s + (6.88 − 11.9i)4-s + (−9.89 − 5.71i)5-s + (20.2 − 13.3i)6-s + (6.90 − 3.98i)7-s − 26.9i·8-s + (26.8 − 3.20i)9-s − 53.3·10-s + (−52.8 + 30.5i)11-s + (32.0 − 64.0i)12-s + (46.4 + 6.45i)13-s + (18.6 − 32.2i)14-s + (−53.0 − 26.5i)15-s + (−7.81 − 13.5i)16-s + 63.8·17-s + ⋯
L(s)  = 1  + (1.42 − 0.824i)2-s + (0.998 − 0.0594i)3-s + (0.861 − 1.49i)4-s + (−0.884 − 0.510i)5-s + (1.37 − 0.908i)6-s + (0.372 − 0.215i)7-s − 1.19i·8-s + (0.992 − 0.118i)9-s − 1.68·10-s + (−1.44 + 0.837i)11-s + (0.771 − 1.54i)12-s + (0.990 + 0.137i)13-s + (0.355 − 0.615i)14-s + (−0.913 − 0.457i)15-s + (−0.122 − 0.211i)16-s + 0.910·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.154 + 0.987i$
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: \((2,\ 117,\ (\ :3/2),\ 0.154 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.07096 - 2.62746i\)
\(L(\frac12)\) \(\approx\) \(3.07096 - 2.62746i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.18 + 0.308i)T \)
13 \( 1 + (-46.4 - 6.45i)T \)
good2 \( 1 + (-4.04 + 2.33i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (9.89 + 5.71i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-6.90 + 3.98i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (52.8 - 30.5i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 63.8T + 4.91e3T^{2} \)
19 \( 1 + 6.27iT - 6.85e3T^{2} \)
23 \( 1 + (83.0 - 143. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-75.2 - 130. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (170. + 98.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 176. iT - 5.06e4T^{2} \)
41 \( 1 + (75.6 + 43.7i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (195. + 339. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (210. - 121. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 99.8T + 1.48e5T^{2} \)
59 \( 1 + (-143. - 82.8i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (11.9 + 20.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-571. - 329. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 818. iT - 3.57e5T^{2} \)
73 \( 1 - 828. iT - 3.89e5T^{2} \)
79 \( 1 + (468. + 811. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (159. - 92.0i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 18.2iT - 7.04e5T^{2} \)
97 \( 1 + (-535. + 309. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.87092186058924791181605766364, −12.16125510439129299182843224809, −11.01650176141938015221136668858, −9.952253846911625231322152565334, −8.325281954909313005350013397325, −7.45351038656682147090369043197, −5.39232941762413615561030862650, −4.25203230207713283982454199837, −3.32955841973916733696052185786, −1.76162131106245510413502245676, 2.93431450570755897016936496428, 3.79016961800206342900649135109, 5.15564492194088297370532190807, 6.53261441285057383159769472507, 7.959470853105617661607841392156, 8.170718085735894559856952273895, 10.29433425224983537310490759797, 11.52422534872919306184208150173, 12.74540348056662293976030854150, 13.50967181991111107715158362152

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