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Properties

Label 2-117-13.9-c3-0-8
Degree $2$
Conductor $117$
Sign $0.525 + 0.850i$
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  + (−1.84 + 3.19i)2-s + (−2.81 − 4.87i)4-s − 18.7·5-s + (12.0 + 20.9i)7-s − 8.74·8-s + (34.6 − 59.9i)10-s + (24.6 − 42.6i)11-s + (−34.3 − 31.9i)13-s − 89.1·14-s + (38.6 − 66.9i)16-s + (32.6 + 56.5i)17-s + (−54.9 − 95.2i)19-s + (52.8 + 91.4i)20-s + (90.9 + 157. i)22-s + (41.6 − 72.1i)23-s + ⋯
L(s)  = 1  + (−0.652 + 1.13i)2-s + (−0.351 − 0.609i)4-s − 1.67·5-s + (0.651 + 1.12i)7-s − 0.386·8-s + (1.09 − 1.89i)10-s + (0.675 − 1.16i)11-s + (−0.732 − 0.681i)13-s − 1.70·14-s + (0.604 − 1.04i)16-s + (0.465 + 0.806i)17-s + (−0.663 − 1.14i)19-s + (0.590 + 1.02i)20-s + (0.881 + 1.52i)22-s + (0.377 − 0.653i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.525 + 0.850i$
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: \((2,\ 117,\ (\ :3/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.181345 - 0.101166i\)
\(L(\frac12)\) \(\approx\) \(0.181345 - 0.101166i\)
\(L(\frac{5}{2})\) 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 \( 1 + (34.3 + 31.9i)T \)
good2 \( 1 + (1.84 - 3.19i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 18.7T + 125T^{2} \)
7 \( 1 + (-12.0 - 20.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.6 + 42.6i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-32.6 - 56.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (54.9 + 95.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-41.6 + 72.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (2.49 - 4.32i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 255.T + 2.97e4T^{2} \)
37 \( 1 + (46.8 - 81.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (33.9 - 58.8i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (71.2 + 123. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 379.T + 1.03e5T^{2} \)
53 \( 1 + 389.T + 1.48e5T^{2} \)
59 \( 1 + (66.9 + 115. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (310. + 537. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (59.5 - 103. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (180. + 312. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 748.T + 3.89e5T^{2} \)
79 \( 1 - 514.T + 4.93e5T^{2} \)
83 \( 1 - 260.T + 5.71e5T^{2} \)
89 \( 1 + (-416. + 721. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-740. - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.63443482496923679808327788001, −11.80834910723134218116400100575, −10.95713466227846847887098461593, −8.996077524886698104718737796932, −8.399074341992388847764838657127, −7.65742560744093870670613099684, −6.40736209718752548193359790745, −5.05016151826977489165173401700, −3.26790258654181164873678307969, −0.14279347358145494278169914010, 1.53152520791604854926011197078, 3.61132804842118143025841401728, 4.52897544215862690158713926295, 7.12200889791737551906448011128, 7.81182541600651102605555734074, 9.188255817321969553235772118448, 10.25081566246862771004317487771, 11.25869338557766727912089449050, 11.86577410524284103784511912027, 12.61449522264245228509781276835

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