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Properties

Label 2-308-7.4-c3-0-16
Degree 22
Conductor 308308
Sign −0.566+0.824i-0.566 + 0.824i
Analytic cond. 18.172518.1725
Root an. cond. 4.262934.26293
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  + (−1.64 + 2.84i)3-s + (−10.0 − 17.4i)5-s + (15.8 + 9.49i)7-s + (8.10 + 14.0i)9-s + (−5.5 + 9.52i)11-s + 18.0·13-s + 66.2·15-s + (−3.37 + 5.83i)17-s + (−79.1 − 137. i)19-s + (−53.1 + 29.6i)21-s + (−42.5 − 73.6i)23-s + (−140. + 244. i)25-s − 141.·27-s − 49.3·29-s + (74.7 − 129. i)31-s + ⋯
L(s)  = 1  + (−0.316 + 0.547i)3-s + (−0.901 − 1.56i)5-s + (0.858 + 0.512i)7-s + (0.300 + 0.519i)9-s + (−0.150 + 0.261i)11-s + 0.384·13-s + 1.14·15-s + (−0.0480 + 0.0832i)17-s + (−0.955 − 1.65i)19-s + (−0.552 + 0.307i)21-s + (−0.385 − 0.667i)23-s + (−1.12 + 1.95i)25-s − 1.01·27-s − 0.316·29-s + (0.433 − 0.750i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.566+0.824i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+3/2) L(s)=((−0.566+0.824i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.566+0.824i-0.566 + 0.824i
Analytic conductor: 18.172518.1725
Root analytic conductor: 4.262934.26293
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ308(221,⋅)\chi_{308} (221, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :3/2), −0.566+0.824i)(2,\ 308,\ (\ :3/2),\ -0.566 + 0.824i)

Particular Values

L(2)L(2) ≈\approx 0.71826026940.7182602694
L(12)L(\frac12) ≈\approx 0.71826026940.7182602694
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
7 1+(−15.8−9.49i)T 1 + (-15.8 - 9.49i)T
11 1+(5.5−9.52i)T 1 + (5.5 - 9.52i)T
good3 1+(1.64−2.84i)T+(−13.5−23.3i)T2 1 + (1.64 - 2.84i)T + (-13.5 - 23.3i)T^{2}
5 1+(10.0+17.4i)T+(−62.5+108.i)T2 1 + (10.0 + 17.4i)T + (-62.5 + 108. i)T^{2}
13 1−18.0T+2.19e3T2 1 - 18.0T + 2.19e3T^{2}
17 1+(3.37−5.83i)T+(−2.45e3−4.25e3i)T2 1 + (3.37 - 5.83i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(79.1+137.i)T+(−3.42e3+5.94e3i)T2 1 + (79.1 + 137. i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(42.5+73.6i)T+(−6.08e3+1.05e4i)T2 1 + (42.5 + 73.6i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+49.3T+2.43e4T2 1 + 49.3T + 2.43e4T^{2}
31 1+(−74.7+129.i)T+(−1.48e4−2.57e4i)T2 1 + (-74.7 + 129. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(142.+246.i)T+(−2.53e4+4.38e4i)T2 1 + (142. + 246. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+223.T+6.89e4T2 1 + 223.T + 6.89e4T^{2}
43 1+38.0T+7.95e4T2 1 + 38.0T + 7.95e4T^{2}
47 1+(205.+355.i)T+(−5.19e4+8.99e4i)T2 1 + (205. + 355. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−178.+309.i)T+(−7.44e4−1.28e5i)T2 1 + (-178. + 309. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−374.+647.i)T+(−1.02e5−1.77e5i)T2 1 + (-374. + 647. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−62.5−108.i)T+(−1.13e5+1.96e5i)T2 1 + (-62.5 - 108. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(318.−552.i)T+(−1.50e5−2.60e5i)T2 1 + (318. - 552. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+176.T+3.57e5T2 1 + 176.T + 3.57e5T^{2}
73 1+(526.−912.i)T+(−1.94e5−3.36e5i)T2 1 + (526. - 912. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(209.+363.i)T+(−2.46e5+4.26e5i)T2 1 + (209. + 363. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+1.47e3T+5.71e5T2 1 + 1.47e3T + 5.71e5T^{2}
89 1+(671.+1.16e3i)T+(−3.52e5+6.10e5i)T2 1 + (671. + 1.16e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−1.28e3T+9.12e5T2 1 - 1.28e3T + 9.12e5T^{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

−11.21109792475573018754807819648, −10.02077000342174524706854054578, −8.711797643467720210598921400168, −8.439345255179660065494956090141, −7.24947784241962891782023151590, −5.50666432525965895334548466707, −4.73335592894751859716707161286, −4.14726877480531587492696732148, −1.95048220957497969563436854669, −0.27963632265759171033911880686, 1.55792732092805029998163457571, 3.31192764581667646214187448450, 4.19300246788901096284359532530, 5.97668132775788877591964023319, 6.82718328442729286872384317509, 7.62466269490293310229623289331, 8.361930241918033591813977924928, 10.13693187619668251399261003479, 10.72417372729609724655431748761, 11.63483574366011021863439710382

Graph of the ZZ-function along the critical line