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Properties

Label 2-108-9.7-c3-0-1
Degree 22
Conductor 108108
Sign 0.655−0.754i0.655 - 0.754i
Analytic cond. 6.372206.37220
Root an. cond. 2.524322.52432
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.37 + 11.0i)5-s + (7.02 − 12.1i)7-s + (−21.2 + 36.8i)11-s + (36.2 + 62.7i)13-s + 59.6·17-s + 105.·19-s + (−0.112 − 0.195i)23-s + (−18.6 + 32.3i)25-s + (−112. + 195. i)29-s + (−100. − 174. i)31-s + 179.·35-s − 152.·37-s + (−244. − 424. i)41-s + (3.79 − 6.57i)43-s + (186. − 323. i)47-s + ⋯
L(s)  = 1  + (0.569 + 0.986i)5-s + (0.379 − 0.657i)7-s + (−0.583 + 1.01i)11-s + (0.772 + 1.33i)13-s + 0.850·17-s + 1.27·19-s + (−0.00102 − 0.00176i)23-s + (−0.149 + 0.258i)25-s + (−0.722 + 1.25i)29-s + (−0.582 − 1.00i)31-s + 0.864·35-s − 0.679·37-s + (−0.932 − 1.61i)41-s + (0.0134 − 0.0233i)43-s + (0.579 − 1.00i)47-s + ⋯

Functional equation

Λ(s)=(108s/2 ΓC(s) L(s)=((0.655−0.754i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(108s/2 ΓC(s+3/2) L(s)=((0.655−0.754i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 108108    =    22⋅332^{2} \cdot 3^{3}
Sign: 0.655−0.754i0.655 - 0.754i
Analytic conductor: 6.372206.37220
Root analytic conductor: 2.524322.52432
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ108(73,⋅)\chi_{108} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 108, ( :3/2), 0.655−0.754i)(2,\ 108,\ (\ :3/2),\ 0.655 - 0.754i)

Particular Values

L(2)L(2) ≈\approx 1.60495+0.731556i1.60495 + 0.731556i
L(12)L(\frac12) ≈\approx 1.60495+0.731556i1.60495 + 0.731556i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
good5 1+(−6.37−11.0i)T+(−62.5+108.i)T2 1 + (-6.37 - 11.0i)T + (-62.5 + 108. i)T^{2}
7 1+(−7.02+12.1i)T+(−171.5−297.i)T2 1 + (-7.02 + 12.1i)T + (-171.5 - 297. i)T^{2}
11 1+(21.2−36.8i)T+(−665.5−1.15e3i)T2 1 + (21.2 - 36.8i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(−36.2−62.7i)T+(−1.09e3+1.90e3i)T2 1 + (-36.2 - 62.7i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1−59.6T+4.91e3T2 1 - 59.6T + 4.91e3T^{2}
19 1−105.T+6.85e3T2 1 - 105.T + 6.85e3T^{2}
23 1+(0.112+0.195i)T+(−6.08e3+1.05e4i)T2 1 + (0.112 + 0.195i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(112.−195.i)T+(−1.21e4−2.11e4i)T2 1 + (112. - 195. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(100.+174.i)T+(−1.48e4+2.57e4i)T2 1 + (100. + 174. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+152.T+5.06e4T2 1 + 152.T + 5.06e4T^{2}
41 1+(244.+424.i)T+(−3.44e4+5.96e4i)T2 1 + (244. + 424. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−3.79+6.57i)T+(−3.97e4−6.88e4i)T2 1 + (-3.79 + 6.57i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(−186.+323.i)T+(−5.19e4−8.99e4i)T2 1 + (-186. + 323. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−43.6T+1.48e5T2 1 - 43.6T + 1.48e5T^{2}
59 1+(335.+581.i)T+(−1.02e5+1.77e5i)T2 1 + (335. + 581. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(37.0−64.1i)T+(−1.13e5−1.96e5i)T2 1 + (37.0 - 64.1i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(210.+364.i)T+(−1.50e5+2.60e5i)T2 1 + (210. + 364. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−730.T+3.57e5T2 1 - 730.T + 3.57e5T^{2}
73 1−473.T+3.89e5T2 1 - 473.T + 3.89e5T^{2}
79 1+(−264.+458.i)T+(−2.46e5−4.26e5i)T2 1 + (-264. + 458. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(−13.0+22.6i)T+(−2.85e5−4.95e5i)T2 1 + (-13.0 + 22.6i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+415.T+7.04e5T2 1 + 415.T + 7.04e5T^{2}
97 1+(463.−803.i)T+(−4.56e5−7.90e5i)T2 1 + (463. - 803. i)T + (-4.56e5 - 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.72088754083852358574208142592, −12.27317025320802554110111636331, −11.08986189468597051632275766481, −10.27227617643410162479026555347, −9.258011435687718671179992135486, −7.56331363203615970169740117166, −6.80310485693927456821688830641, −5.29664639856771086251717932999, −3.65371884616420700930501210720, −1.85921228953173603111288956228, 1.12322978972180077391822437651, 3.16690435528968200793987770574, 5.29545941711378763869726626505, 5.74365818668759514831955687625, 7.86887695385306704855140977479, 8.652897468187192026073441277333, 9.780917802161537685562290029172, 11.00006466291121156507867643897, 12.15359864213311094242685225230, 13.14153035229829597259143982924

Graph of the ZZ-function along the critical line