Properties

Label 2-117-13.10-c3-0-2
Degree $2$
Conductor $117$
Sign $-0.711 - 0.702i$
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  + (−4 + 6.92i)4-s − 5.19i·5-s + (9 + 5.19i)7-s + (−45 + 25.9i)11-s + (−32.5 + 33.7i)13-s + (−31.9 − 55.4i)16-s + (−58.5 + 101. i)17-s + (−21 − 12.1i)19-s + (36 + 20.7i)20-s + (9 + 15.5i)23-s + 98·25-s + (−72 + 41.5i)28-s + (−49.5 − 85.7i)29-s + 193. i·31-s + (27 − 46.7i)35-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s − 0.464i·5-s + (0.485 + 0.280i)7-s + (−1.23 + 0.712i)11-s + (−0.693 + 0.720i)13-s + (−0.499 − 0.866i)16-s + (−0.834 + 1.44i)17-s + (−0.253 − 0.146i)19-s + (0.402 + 0.232i)20-s + (0.0815 + 0.141i)23-s + 0.784·25-s + (−0.485 + 0.280i)28-s + (−0.316 − 0.548i)29-s + 1.12i·31-s + (0.130 − 0.225i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.311260 + 0.758315i\)
\(L(\frac12)\) \(\approx\) \(0.311260 + 0.758315i\)
\(L(\frac{5}{2})\) 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 \( 1 + (32.5 - 33.7i)T \)
good2 \( 1 + (4 - 6.92i)T^{2} \)
5 \( 1 + 5.19iT - 125T^{2} \)
7 \( 1 + (-9 - 5.19i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (45 - 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (58.5 - 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21 + 12.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-9 - 15.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (49.5 + 85.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 193. iT - 2.97e4T^{2} \)
37 \( 1 + (-97.5 + 56.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-31.5 + 18.1i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-41 + 71.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 72.7iT - 1.03e5T^{2} \)
53 \( 1 - 261T + 1.48e5T^{2} \)
59 \( 1 + (-684 - 394. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-359.5 + 622. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (609 - 351. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-405 - 233. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 684. iT - 3.89e5T^{2} \)
79 \( 1 + 440T + 4.93e5T^{2} \)
83 \( 1 + 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 + (1.31e3 - 758. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.00e3 + 578. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.05771874583364804611106677908, −12.68064059055099393617718597957, −11.55045472453187082248003071398, −10.24712723831116510034059204074, −8.937311633364286390907241895022, −8.188425080815976218225060093168, −7.04690718331938845595254147137, −5.16005717873446258472430934273, −4.22199163034342550816395719015, −2.30767924431554354842679496555, 0.42500067284976818809626589803, 2.65474385746469661570119724731, 4.68499724603489142507592209860, 5.62911058220023515251421581793, 7.12709172042544297898385412472, 8.344996066261117028696859587830, 9.627790666267873245272888912027, 10.62127225515077707423305950280, 11.27452949265744202545108335743, 12.95394563306508863529852704814

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