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Properties

Label 2-117-13.10-c3-0-12
Degree $2$
Conductor $117$
Sign $-0.836 + 0.547i$
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  + (1.06 − 0.614i)2-s + (−3.24 + 5.62i)4-s − 17.3i·5-s + (−16.6 − 9.59i)7-s + 17.8i·8-s + (−10.6 − 18.4i)10-s + (−28.2 + 16.3i)11-s + (−26.1 − 38.8i)13-s − 23.5·14-s + (−15.0 − 26.0i)16-s + (43.2 − 74.9i)17-s + (−65.8 − 38.0i)19-s + (97.2 + 56.1i)20-s + (−20.0 + 34.7i)22-s + (−37.6 − 65.2i)23-s + ⋯
L(s)  = 1  + (0.376 − 0.217i)2-s + (−0.405 + 0.702i)4-s − 1.54i·5-s + (−0.897 − 0.518i)7-s + 0.786i·8-s + (−0.336 − 0.582i)10-s + (−0.775 + 0.447i)11-s + (−0.557 − 0.829i)13-s − 0.450·14-s + (−0.234 − 0.406i)16-s + (0.617 − 1.06i)17-s + (−0.795 − 0.459i)19-s + (1.08 + 0.627i)20-s + (−0.194 + 0.336i)22-s + (−0.341 − 0.591i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.836 + 0.547i$
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: \((2,\ 117,\ (\ :3/2),\ -0.836 + 0.547i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.235178 - 0.789454i\)
\(L(\frac12)\) \(\approx\) \(0.235178 - 0.789454i\)
\(L(\frac{5}{2})\) 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 \( 1 + (26.1 + 38.8i)T \)
good2 \( 1 + (-1.06 + 0.614i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 17.3iT - 125T^{2} \)
7 \( 1 + (16.6 + 9.59i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (28.2 - 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-43.2 + 74.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (65.8 + 38.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (37.6 + 65.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-143. - 248. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 200. iT - 2.97e4T^{2} \)
37 \( 1 + (-149. + 86.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-429. + 248. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (93.7 - 162. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 254. iT - 1.03e5T^{2} \)
53 \( 1 + 63.6T + 1.48e5T^{2} \)
59 \( 1 + (58.8 + 33.9i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (200. - 347. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-442. + 255. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (173. + 100. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 168. iT - 3.89e5T^{2} \)
79 \( 1 + 1.11e3T + 4.93e5T^{2} \)
83 \( 1 + 724. iT - 5.71e5T^{2} \)
89 \( 1 + (-711. + 410. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (16.9 + 9.78i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.68333809392702323137885828988, −12.19065192355571252194359242220, −10.46651664793609859560038975729, −9.321638992541372510313096824183, −8.372471447020899275749281170109, −7.27394545194141799611074055653, −5.27978514533831332647977232122, −4.49215935864131242730984820140, −2.92575761076214194650279068561, −0.37592473692954580509090685607, 2.56907585091108900035883279565, 4.04213175947267956314969074388, 5.96329748937282302133798558155, 6.37423399397580105300973370985, 7.87559429524321679297787902536, 9.610867754476645188960619328934, 10.19773329861018297113781769887, 11.26845594101521030763802883348, 12.65710175383918644099252653578, 13.66049211727428453660832926691

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