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Properties

Label 2-117-13.12-c3-0-5
Degree $2$
Conductor $117$
Sign $-0.159 - 0.987i$
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  + 1.32i·2-s + 6.23·4-s + 15.4i·5-s − 7.96i·7-s + 18.9i·8-s − 20.4·10-s − 12.7i·11-s + (7.47 + 46.2i)13-s + 10.5·14-s + 24.8·16-s − 54·17-s + 84.5i·19-s + 96.2i·20-s + 16.9·22-s + 122.·23-s + ⋯
L(s)  = 1  + 0.469i·2-s + 0.779·4-s + 1.37i·5-s − 0.430i·7-s + 0.835i·8-s − 0.647·10-s − 0.350i·11-s + (0.159 + 0.987i)13-s + 0.201·14-s + 0.387·16-s − 0.770·17-s + 1.02i·19-s + 1.07i·20-s + 0.164·22-s + 1.11·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20047 + 1.40998i\)
\(L(\frac12)\) \(\approx\) \(1.20047 + 1.40998i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.47 - 46.2i)T \)
good2 \( 1 - 1.32iT - 8T^{2} \)
5 \( 1 - 15.4iT - 125T^{2} \)
7 \( 1 + 7.96iT - 343T^{2} \)
11 \( 1 + 12.7iT - 1.33e3T^{2} \)
17 \( 1 + 54T + 4.91e3T^{2} \)
19 \( 1 - 84.5iT - 6.85e3T^{2} \)
23 \( 1 - 122.T + 1.21e4T^{2} \)
29 \( 1 + 140.T + 2.43e4T^{2} \)
31 \( 1 - 116. iT - 2.97e4T^{2} \)
37 \( 1 + 433. iT - 5.06e4T^{2} \)
41 \( 1 + 205. iT - 6.89e4T^{2} \)
43 \( 1 - 418.T + 7.95e4T^{2} \)
47 \( 1 + 485. iT - 1.03e5T^{2} \)
53 \( 1 - 674.T + 1.48e5T^{2} \)
59 \( 1 + 186. iT - 2.05e5T^{2} \)
61 \( 1 + 671.T + 2.26e5T^{2} \)
67 \( 1 + 14.0iT - 3.00e5T^{2} \)
71 \( 1 + 346. iT - 3.57e5T^{2} \)
73 \( 1 + 832. iT - 3.89e5T^{2} \)
79 \( 1 + 335.T + 4.93e5T^{2} \)
83 \( 1 - 568. iT - 5.71e5T^{2} \)
89 \( 1 - 236. iT - 7.04e5T^{2} \)
97 \( 1 - 1.27e3iT - 9.12e5T^{2} \)
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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

−13.70101119832768932846734489477, −12.10794185675480091614875837619, −10.95480501464025518972256874935, −10.65005991566131917565059133257, −8.970739713369947696623887366944, −7.40912475143256646982549569631, −6.87213067379417310450204450260, −5.78661490425240969965153329397, −3.70510994033716105615211728630, −2.21358440604302909228279161300, 1.04219626030381770724059870583, 2.70537249330480534237819096703, 4.55803331966515885335514194982, 5.83292012781761417997701667359, 7.30500279823175762453336620302, 8.615967190490185903399101696328, 9.554563662502972537838976732652, 10.86609103508001420155657237941, 11.76977748032808340142275505743, 12.80471218380316483236689017640

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