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Properties

Label 2-117-13.4-c3-0-14
Degree $2$
Conductor $117$
Sign $-0.474 + 0.880i$
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  + (1.76 + 1.02i)2-s + (−1.91 − 3.31i)4-s − 12.0i·5-s + (−25.7 + 14.8i)7-s − 24.1i·8-s + (12.3 − 21.3i)10-s + (−24.3 − 14.0i)11-s + (−40.9 − 22.7i)13-s − 60.7·14-s + (9.35 − 16.1i)16-s + (25.3 + 43.8i)17-s + (91.0 − 52.5i)19-s + (−40.0 + 23.1i)20-s + (−28.6 − 49.6i)22-s + (80.2 − 139. i)23-s + ⋯
L(s)  = 1  + (0.625 + 0.361i)2-s + (−0.239 − 0.414i)4-s − 1.08i·5-s + (−1.39 + 0.802i)7-s − 1.06i·8-s + (0.390 − 0.675i)10-s + (−0.666 − 0.384i)11-s + (−0.874 − 0.485i)13-s − 1.15·14-s + (0.146 − 0.253i)16-s + (0.361 + 0.625i)17-s + (1.09 − 0.634i)19-s + (−0.447 + 0.258i)20-s + (−0.277 − 0.481i)22-s + (0.727 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.474 + 0.880i$
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: \((2,\ 117,\ (\ :3/2),\ -0.474 + 0.880i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.575366 - 0.963747i\)
\(L(\frac12)\) \(\approx\) \(0.575366 - 0.963747i\)
\(L(\frac{5}{2})\) 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 \( 1 + (40.9 + 22.7i)T \)
good2 \( 1 + (-1.76 - 1.02i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 12.0iT - 125T^{2} \)
7 \( 1 + (25.7 - 14.8i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (24.3 + 14.0i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-25.3 - 43.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-91.0 + 52.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-80.2 + 139. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.0 + 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 223. iT - 2.97e4T^{2} \)
37 \( 1 + (197. + 114. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (256. + 147. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-96.0 - 166. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 36.9iT - 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 + (-380. + 219. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (143. + 247. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-465. - 268. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (88.9 - 51.3i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 75.5iT - 3.89e5T^{2} \)
79 \( 1 - 17.5T + 4.93e5T^{2} \)
83 \( 1 + 1.46e3iT - 5.71e5T^{2} \)
89 \( 1 + (-290. - 167. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (648. - 374. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.71157134844501180457958732450, −12.33619519046962230374153223827, −10.36791843089031200521059480323, −9.478452443245323825096396226232, −8.553652209447799212090777953025, −6.82838810078714819470367151904, −5.61170686266666342507983786341, −4.89425248456744510990677072628, −3.09591827221167367968080485903, −0.47927265435100376165508887381, 2.82662655966447876440912990499, 3.60557315618459288301773714746, 5.21433305813647267221827924610, 6.91349672711147413413736306687, 7.58186621036190472882584015716, 9.496585329723919222966880794739, 10.25915010998192197344143059387, 11.50148400307945603774519111418, 12.45368779497373591756906502619, 13.46669493112945568932115562187

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