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Properties

Label 2-117-117.103-c3-0-10
Degree $2$
Conductor $117$
Sign $0.262 - 0.964i$
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  + (−3.79 + 2.19i)2-s + (−4.80 + 1.97i)3-s + (5.60 − 9.70i)4-s + (−14.9 − 8.64i)5-s + (13.9 − 18.0i)6-s + (−3.77 + 2.17i)7-s + 14.0i·8-s + (19.2 − 18.9i)9-s + 75.8·10-s + (−15.4 + 8.94i)11-s + (−7.77 + 57.7i)12-s + (−16.1 − 44.0i)13-s + (9.54 − 16.5i)14-s + (89.0 + 11.9i)15-s + (13.9 + 24.2i)16-s − 87.8·17-s + ⋯
L(s)  = 1  + (−1.34 + 0.774i)2-s + (−0.924 + 0.380i)3-s + (0.700 − 1.21i)4-s + (−1.33 − 0.773i)5-s + (0.946 − 1.22i)6-s + (−0.203 + 0.117i)7-s + 0.622i·8-s + (0.711 − 0.703i)9-s + 2.39·10-s + (−0.424 + 0.245i)11-s + (−0.186 + 1.38i)12-s + (−0.343 − 0.939i)13-s + (0.182 − 0.315i)14-s + (1.53 + 0.206i)15-s + (0.218 + 0.378i)16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.262 - 0.964i$
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: \((2,\ 117,\ (\ :3/2),\ 0.262 - 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.191646 + 0.146483i\)
\(L(\frac12)\) \(\approx\) \(0.191646 + 0.146483i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.80 - 1.97i)T \)
13 \( 1 + (16.1 + 44.0i)T \)
good2 \( 1 + (3.79 - 2.19i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (14.9 + 8.64i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (3.77 - 2.17i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (15.4 - 8.94i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + 87.8T + 4.91e3T^{2} \)
19 \( 1 - 63.3iT - 6.85e3T^{2} \)
23 \( 1 + (22.2 - 38.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-13.8 - 24.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-206. - 119. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 258. iT - 5.06e4T^{2} \)
41 \( 1 + (-252. - 145. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (152. + 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-265. + 153. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 696.T + 1.48e5T^{2} \)
59 \( 1 + (-185. - 106. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (286. + 496. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-541. - 312. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 190. iT - 3.57e5T^{2} \)
73 \( 1 - 734. iT - 3.89e5T^{2} \)
79 \( 1 + (254. + 441. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (327. - 188. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 580. iT - 7.04e5T^{2} \)
97 \( 1 + (871. - 502. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.86658930021399489110753721438, −12.08627459110704452199088239516, −10.90537792110283625339549818322, −10.00253993323132093100196210960, −8.830902072906160679076892895803, −7.909801457312262519548449076740, −6.93202012716711198895787517849, −5.51642172762921995221294380889, −4.13101300541194405590855061496, −0.63596334687436169800074155694, 0.42665619796654500131056675990, 2.51498242324858488494194670634, 4.42329162623970699996415297916, 6.62650298547092134839551362999, 7.46845824507629414975043773652, 8.519529791342475674932019828034, 9.949007664142777098896413658773, 10.97071963779518694372639277152, 11.43481352706375699579211425933, 12.16703500124698101805573249568

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