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Properties

Label 2-117-13.9-c3-0-4
Degree $2$
Conductor $117$
Sign $0.216 - 0.976i$
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  + (−0.643 + 1.11i)2-s + (3.17 + 5.49i)4-s + 12.4·5-s + (4.50 + 7.79i)7-s − 18.4·8-s + (−8.00 + 13.8i)10-s + (25.5 − 44.2i)11-s + (2.33 + 46.8i)13-s − 11.5·14-s + (−13.5 + 23.3i)16-s + (3.43 + 5.94i)17-s + (41.5 + 71.9i)19-s + (39.4 + 68.3i)20-s + (32.8 + 56.9i)22-s + (−93.7 + 162. i)23-s + ⋯
L(s)  = 1  + (−0.227 + 0.393i)2-s + (0.396 + 0.686i)4-s + 1.11·5-s + (0.243 + 0.420i)7-s − 0.815·8-s + (−0.253 + 0.438i)10-s + (0.700 − 1.21i)11-s + (0.0498 + 0.998i)13-s − 0.221·14-s + (−0.210 + 0.365i)16-s + (0.0489 + 0.0847i)17-s + (0.501 + 0.869i)19-s + (0.441 + 0.764i)20-s + (0.318 + 0.551i)22-s + (−0.850 + 1.47i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.216 - 0.976i$
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: \((2,\ 117,\ (\ :3/2),\ 0.216 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.44969 + 1.16328i\)
\(L(\frac12)\) \(\approx\) \(1.44969 + 1.16328i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.33 - 46.8i)T \)
good2 \( 1 + (0.643 - 1.11i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 12.4T + 125T^{2} \)
7 \( 1 + (-4.50 - 7.79i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-25.5 + 44.2i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-3.43 - 5.94i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-41.5 - 71.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (93.7 - 162. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-111. + 193. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 57.3T + 2.97e4T^{2} \)
37 \( 1 + (-78.1 + 135. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (111. - 192. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (173. + 301. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 45.0T + 1.03e5T^{2} \)
53 \( 1 - 473.T + 1.48e5T^{2} \)
59 \( 1 + (307. + 533. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (97.6 + 169. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-177. + 307. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (381. + 661. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 331.T + 3.89e5T^{2} \)
79 \( 1 + 207.T + 4.93e5T^{2} \)
83 \( 1 + 251.T + 5.71e5T^{2} \)
89 \( 1 + (359. - 622. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-778. - 1.34e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.55424744885595447607405272882, −12.00684142074328726550860432557, −11.48143720145887269376717496682, −9.856036222482208679445192187315, −8.932265377478680634896783283756, −7.922463245319964783844773399716, −6.44696618454946839331007380240, −5.74153221395526026160677458209, −3.62959414289594633435130607490, −1.94622504409769763197675964336, 1.21052035283430772864006123439, 2.57010953556959715715978230001, 4.80526132761550649263740189896, 6.08076058811943495992630505543, 7.10765787827366949623668394202, 8.856135418442609066233895557397, 10.02687386826816716281036582641, 10.35839342162186161221922797234, 11.70812452764659516991110875197, 12.75227438978438365448857763872

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