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Properties

Label 2-117-117.103-c3-0-26
Degree $2$
Conductor $117$
Sign $0.970 - 0.241i$
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  + (1.88 − 1.08i)2-s + (5.16 − 0.539i)3-s + (−1.63 + 2.82i)4-s + (12.6 + 7.27i)5-s + (9.15 − 6.64i)6-s + (−9.24 + 5.33i)7-s + 24.5i·8-s + (26.4 − 5.57i)9-s + 31.6·10-s + (−23.1 + 13.3i)11-s + (−6.90 + 15.4i)12-s + (6.45 − 46.4i)13-s + (−11.6 + 20.1i)14-s + (69.0 + 30.8i)15-s + (13.6 + 23.6i)16-s − 10.9·17-s + ⋯
L(s)  = 1  + (0.666 − 0.384i)2-s + (0.994 − 0.103i)3-s + (−0.203 + 0.353i)4-s + (1.12 + 0.651i)5-s + (0.622 − 0.451i)6-s + (−0.499 + 0.288i)7-s + 1.08i·8-s + (0.978 − 0.206i)9-s + 1.00·10-s + (−0.634 + 0.366i)11-s + (−0.166 + 0.372i)12-s + (0.137 − 0.990i)13-s + (−0.221 + 0.384i)14-s + (1.18 + 0.530i)15-s + (0.213 + 0.368i)16-s − 0.156·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.970 - 0.241i$
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: \((2,\ 117,\ (\ :3/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.10597 + 0.381309i\)
\(L(\frac12)\) \(\approx\) \(3.10597 + 0.381309i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.16 + 0.539i)T \)
13 \( 1 + (-6.45 + 46.4i)T \)
good2 \( 1 + (-1.88 + 1.08i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-12.6 - 7.27i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (9.24 - 5.33i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (23.1 - 13.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + 10.9T + 4.91e3T^{2} \)
19 \( 1 + 62.0iT - 6.85e3T^{2} \)
23 \( 1 + (-97.2 + 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (34.2 + 59.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (83.0 + 47.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 184. iT - 5.06e4T^{2} \)
41 \( 1 + (-271. - 156. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-0.207 - 0.358i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-30.0 + 17.3i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 638.T + 1.48e5T^{2} \)
59 \( 1 + (307. + 177. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-18.6 - 32.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (431. + 249. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 151. iT - 3.57e5T^{2} \)
73 \( 1 - 1.02e3iT - 3.89e5T^{2} \)
79 \( 1 + (-556. - 964. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (46.4 - 26.8i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + (-1.41e3 + 819. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.90585974877726903476674476207, −12.82231104262582649985253588007, −10.94424000254587520361128762902, −9.912662954123377850434788358375, −8.887010900427513822126108866159, −7.70596230215531322139811270717, −6.29235897868869384136885603567, −4.77953826568634347022534843389, −3.05367711849280482315115210858, −2.43444917784070834409149396927, 1.60765873557103796737595542854, 3.57496818106357037866915292338, 4.96079605511230651096025333939, 6.06151509139343627495150972594, 7.38345219535829729141997181861, 9.099463215083998977906762588563, 9.489270705017616784680676458857, 10.61339416535315096025996517201, 12.66770141062060238418008488065, 13.37381955384781326920687860790

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