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Properties

Label 2-117-117.103-c3-0-19
Degree $2$
Conductor $117$
Sign $0.123 - 0.992i$
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  + (−1.00 + 0.578i)2-s + (4.54 + 2.51i)3-s + (−3.33 + 5.76i)4-s + (9.98 + 5.76i)5-s + (−6.01 + 0.116i)6-s + (23.3 − 13.5i)7-s − 16.9i·8-s + (14.3 + 22.8i)9-s − 13.3·10-s + (−7.49 + 4.32i)11-s + (−29.6 + 17.8i)12-s + (−35.5 + 30.5i)13-s + (−15.6 + 27.0i)14-s + (30.9 + 51.2i)15-s + (−16.8 − 29.1i)16-s + 30.4·17-s + ⋯
L(s)  = 1  + (−0.354 + 0.204i)2-s + (0.875 + 0.483i)3-s + (−0.416 + 0.720i)4-s + (0.892 + 0.515i)5-s + (−0.409 + 0.00789i)6-s + (1.26 − 0.729i)7-s − 0.749i·8-s + (0.533 + 0.846i)9-s − 0.421·10-s + (−0.205 + 0.118i)11-s + (−0.712 + 0.430i)12-s + (−0.758 + 0.651i)13-s + (−0.298 + 0.516i)14-s + (0.532 + 0.882i)15-s + (−0.262 − 0.455i)16-s + 0.434·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.123 - 0.992i$
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: \((2,\ 117,\ (\ :3/2),\ 0.123 - 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.48635 + 1.31276i\)
\(L(\frac12)\) \(\approx\) \(1.48635 + 1.31276i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.54 - 2.51i)T \)
13 \( 1 + (35.5 - 30.5i)T \)
good2 \( 1 + (1.00 - 0.578i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-9.98 - 5.76i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-23.3 + 13.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (7.49 - 4.32i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 30.4T + 4.91e3T^{2} \)
19 \( 1 - 30.3iT - 6.85e3T^{2} \)
23 \( 1 + (-3.48 + 6.03i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-88.4 - 153. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (272. + 157. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 122. iT - 5.06e4T^{2} \)
41 \( 1 + (-92.3 - 53.3i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (185. + 320. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-246. + 142. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 618.T + 1.48e5T^{2} \)
59 \( 1 + (719. + 415. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (289. + 501. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-554. - 320. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 725. iT - 3.57e5T^{2} \)
73 \( 1 + 5.08iT - 3.89e5T^{2} \)
79 \( 1 + (-405. - 702. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-680. + 392. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 + (-270. + 156. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.72094702960016718163552131833, −12.41680633001463633105325023330, −10.86354156800493522671547832821, −9.941573201303635617053747585581, −9.022351326222979502911323896135, −7.88711770698824586961234547690, −7.15006292619356204017886384829, −5.00054666612799876155619561694, −3.79302823431874476414157568208, −2.09508976026678954702237535836, 1.28964971101156945120404504442, 2.39392053270668120938258466477, 4.88116051755932967480136122745, 5.79866845787140681369384432500, 7.71096684243226321829002584521, 8.696806216269626736070482355435, 9.407014757822078548184718284894, 10.44403722037071751870816803948, 11.85411560113227368572439113759, 12.99719041275131759574581515488

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