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Properties

Label 2-117-117.103-c3-0-28
Degree $2$
Conductor $117$
Sign $0.902 + 0.429i$
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 + (46.1 − 8.04i)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 + (0.985 − 0.171i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.902 + 0.429i$
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: \((2,\ 117,\ (\ :3/2),\ 0.902 + 0.429i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.98424 - 0.673791i\)
\(L(\frac12)\) \(\approx\) \(2.98424 - 0.673791i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.1 + 8.04i)T \)
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

−13.27297642071692099229571310311, −11.83113579734383553195663295639, −10.99231161899988105245693788716, −10.47267415790420232054383688541, −9.163167248899808438317175469134, −6.62664139829730214033067763639, −5.94793129942985457759309597656, −4.85043462111568513681575744587, −3.47487433155230307381852595071, −1.81897423239352257962280261091, 1.67959704375138675434062198185, 4.27058258374211545765815746129, 5.35425993535957508502615800771, 6.08081838961739503156413600694, 6.93211082389680254768274365343, 8.663036011793722797451012629335, 10.08193857312860225683610416155, 11.46568101845071969981926081086, 12.56801549913538593962910344046, 13.21604022383244047655161257738

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