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Properties

Label 2-117-13.4-c3-0-15
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: $\chi_{117} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.836 - 0.547i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0800722 + 0.268789i\)
\(L(\frac12)\) \(\approx\) \(0.0800722 + 0.268789i\)
\(L(\frac{5}{2})\) 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

−12.45861847745865323348950183590, −11.52089449026975099038380593540, −9.925920596955235498269558293579, −9.164130095156342073954125604856, −8.705679171721753408209876788906, −6.75131612021142909668460906514, −5.35800523731409452111554466492, −4.36769726646268844758593472444, −1.85513449892764155251562460182, −0.16552113298127580186628576779, 2.98892733072705219761173581880, 3.94551484475245193424086065461, 6.32985977149749776742760036462, 7.02562090358624111007427265564, 8.162271382610640018649802385528, 9.518833778590974850978090936172, 10.39936600995505099509215002619, 11.43607806826440510520249935877, 12.84321465694359592191205782916, 13.54124300814287197991796223217

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