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Properties

Label 2-117-13.12-c3-0-12
Degree $2$
Conductor $117$
Sign $0.847 + 0.531i$
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  + 1.52i·2-s + 5.67·4-s − 9.65i·5-s − 22.3i·7-s + 20.8i·8-s + 14.7·10-s − 50.3i·11-s + (−39.7 − 24.8i)13-s + 34.0·14-s + 13.6·16-s + 86.1·17-s + 116. i·19-s − 54.8i·20-s + 76.6·22-s + 72·23-s + ⋯
L(s)  = 1  + 0.538i·2-s + 0.709·4-s − 0.863i·5-s − 1.20i·7-s + 0.921i·8-s + 0.465·10-s − 1.37i·11-s + (−0.847 − 0.531i)13-s + 0.650·14-s + 0.213·16-s + 1.22·17-s + 1.41i·19-s − 0.613i·20-s + 0.742·22-s + 0.652·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.847 + 0.531i$
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: \((2,\ 117,\ (\ :3/2),\ 0.847 + 0.531i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.77374 - 0.509953i\)
\(L(\frac12)\) \(\approx\) \(1.77374 - 0.509953i\)
\(L(\frac{5}{2})\) 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 \( 1 + (39.7 + 24.8i)T \)
good2 \( 1 - 1.52iT - 8T^{2} \)
5 \( 1 + 9.65iT - 125T^{2} \)
7 \( 1 + 22.3iT - 343T^{2} \)
11 \( 1 + 50.3iT - 1.33e3T^{2} \)
17 \( 1 - 86.1T + 4.91e3T^{2} \)
19 \( 1 - 116. iT - 6.85e3T^{2} \)
23 \( 1 - 72T + 1.21e4T^{2} \)
29 \( 1 + 14.1T + 2.43e4T^{2} \)
31 \( 1 + 196. iT - 2.97e4T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 - 265. iT - 6.89e4T^{2} \)
43 \( 1 + 211.T + 7.95e4T^{2} \)
47 \( 1 - 67.5iT - 1.03e5T^{2} \)
53 \( 1 + 686.T + 1.48e5T^{2} \)
59 \( 1 + 91.9iT - 2.05e5T^{2} \)
61 \( 1 - 329.T + 2.26e5T^{2} \)
67 \( 1 - 768. iT - 3.00e5T^{2} \)
71 \( 1 + 264. iT - 3.57e5T^{2} \)
73 \( 1 + 771. iT - 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 514. iT - 5.71e5T^{2} \)
89 \( 1 - 527. iT - 7.04e5T^{2} \)
97 \( 1 + 74.2iT - 9.12e5T^{2} \)
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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

−13.01634242258812029277010841719, −11.94519113024874889710124350570, −10.86210088355678574358683892328, −9.865537651948490244749104511366, −8.214150318880025165172328984056, −7.63396048998642862936050256704, −6.20855899534886040772748113890, −5.10651623012588345471476611192, −3.34214115959498038579488493415, −1.05053209099204594923819581562, 2.08189587908556452875941386153, 3.05039960954057654659775715653, 5.07175870306743126644524842705, 6.68149850200305234448488734534, 7.36835945073813770692136581651, 9.192916961285543671668854343802, 10.10935121479745511034532571592, 11.15702994662389014541366348690, 12.12491500963534530172409587462, 12.64133551898142347629947662330

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