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Properties

Label 2-117-117.103-c3-0-12
Degree $2$
Conductor $117$
Sign $-0.522 - 0.852i$
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 + (−8.69 + 46.0i)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 + (−0.185 + 0.982i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.522 - 0.852i$
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: \((2,\ 117,\ (\ :3/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.668042 + 1.19340i\)
\(L(\frac12)\) \(\approx\) \(0.668042 + 1.19340i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.69 - 46.0i)T \)
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

−13.35945028594184878013755124346, −12.37181280303835610274876847420, −11.77290886360700386495420999310, −9.999961208080544061670090357734, −8.889566415510747458470702787142, −8.405752668974612646601462605312, −6.95327136574927171024230369063, −4.89438452590702992911697932404, −3.80906370861836437803706139397, −2.82143039611228303750081207477, 0.59943728296404665785341852992, 3.15797870100497089457082243893, 4.13583573149671157788706387380, 6.15515826238762621994350268010, 7.11950358559005016879744417541, 8.159179724913747751659007644611, 9.715908416514409715827057582703, 10.23297095205345836201772674185, 11.93528107925083166372698110885, 13.11437019662659909694800305262

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