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Properties

Label 2-3332-476.179-c0-0-3
Degree 22
Conductor 33323332
Sign 0.989+0.142i0.989 + 0.142i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.758 + 0.0999i)5-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (−0.758 − 0.0999i)10-s − 2i·13-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)17-s + (0.499 − 0.866i)18-s + (−0.707 − 0.292i)20-s + (−0.400 + 0.107i)25-s + (0.517 − 1.93i)26-s + (0.707 − 1.70i)29-s + (0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.758 + 0.0999i)5-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (−0.758 − 0.0999i)10-s − 2i·13-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)17-s + (0.499 − 0.866i)18-s + (−0.707 − 0.292i)20-s + (−0.400 + 0.107i)25-s + (0.517 − 1.93i)26-s + (0.707 − 1.70i)29-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.989+0.142i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.989+0.142i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.989+0.142i0.989 + 0.142i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(655,⋅)\chi_{3332} (655, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.989+0.142i)(2,\ 3332,\ (\ :0),\ 0.989 + 0.142i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1593513982.159351398
L(12)L(\frac12) ≈\approx 2.1593513982.159351398
L(1)L(1) 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+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
7 1 1
17 1+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
good3 1+(−0.258+0.965i)T2 1 + (-0.258 + 0.965i)T^{2}
5 1+(0.758−0.0999i)T+(0.965−0.258i)T2 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2}
11 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
13 1+2iT−T2 1 + 2iT - T^{2}
19 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
23 1+(−0.258−0.965i)T2 1 + (-0.258 - 0.965i)T^{2}
29 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
31 1+(−0.258+0.965i)T2 1 + (-0.258 + 0.965i)T^{2}
37 1+(−0.241−1.83i)T+(−0.965+0.258i)T2 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2}
41 1+(−0.707−1.70i)T+(−0.707+0.707i)T2 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
59 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
61 1+(1.12−1.46i)T+(−0.258−0.965i)T2 1 + (1.12 - 1.46i)T + (-0.258 - 0.965i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
73 1+(−0.465−0.607i)T+(−0.258+0.965i)T2 1 + (-0.465 - 0.607i)T + (-0.258 + 0.965i)T^{2}
79 1+(−0.258−0.965i)T2 1 + (-0.258 - 0.965i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.292+0.707i)T+(−0.707−0.707i)T2 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)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

−8.296338926915897265031181400278, −7.972728831318144968301114443732, −7.33536404702883153314583440712, −6.33601972275482410971678787166, −5.84688512575265743694388643770, −4.91999534864528773051790066927, −4.11231490799299280002807728559, −3.27012524113220729940719904255, −2.81929553358394672626968431942, −1.05036628784670315363344964580, 1.51601818799201995453862550902, 2.33148123389208182931169255910, 3.54622319367956861352127656597, 4.17605824494261548442777343022, 4.82196630763423510199821965572, 5.63791718991454322497122525005, 6.55556229380696330112643496625, 7.33786990202874253531838727636, 7.74732088415036399044420867513, 8.879790457166942494605430963277

Graph of the ZZ-function along the critical line