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Properties

Label 2-3332-476.151-c0-0-1
Degree 22
Conductor 33323332
Sign 0.867−0.497i0.867 - 0.497i
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.607 − 0.465i)5-s + (0.707 − 0.707i)8-s + (0.965 + 0.258i)9-s + (0.607 − 0.465i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (0.292 + 0.707i)20-s + (−0.107 − 0.400i)25-s + (0.707 − 0.292i)29-s + (−0.965 + 0.258i)32-s − i·34-s + (−0.707 − 0.707i)36-s + (0.465 − 0.607i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.607 − 0.465i)5-s + (0.707 − 0.707i)8-s + (0.965 + 0.258i)9-s + (0.607 − 0.465i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (0.292 + 0.707i)20-s + (−0.107 − 0.400i)25-s + (0.707 − 0.292i)29-s + (−0.965 + 0.258i)32-s − i·34-s + (−0.707 − 0.707i)36-s + (0.465 − 0.607i)37-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.867−0.497i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.867−0.497i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.867−0.497i0.867 - 0.497i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(3007,⋅)\chi_{3332} (3007, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.867−0.497i)(2,\ 3332,\ (\ :0),\ 0.867 - 0.497i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92514186080.9251418608
L(12)L(\frac12) ≈\approx 0.92514186080.9251418608
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+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
7 1 1
17 1+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
good3 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
5 1+(0.607+0.465i)T+(0.258+0.965i)T2 1 + (0.607 + 0.465i)T + (0.258 + 0.965i)T^{2}
11 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
13 1−T2 1 - T^{2}
19 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
23 1+(−0.965+0.258i)T2 1 + (-0.965 + 0.258i)T^{2}
29 1+(−0.707+0.292i)T+(0.707−0.707i)T2 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2}
31 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
37 1+(−0.465+0.607i)T+(−0.258−0.965i)T2 1 + (-0.465 + 0.607i)T + (-0.258 - 0.965i)T^{2}
41 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)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+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
59 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
61 1+(0.241+1.83i)T+(−0.965+0.258i)T2 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)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+(−0.0999+0.758i)T+(−0.965−0.258i)T2 1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2}
79 1+(−0.965+0.258i)T2 1 + (-0.965 + 0.258i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
97 1+(−0.707+0.292i)T+(0.707−0.707i)T2 1 + (-0.707 + 0.292i)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

−8.766593190917687965545589207440, −7.941966004847537077871514734704, −7.55931249176934566100841979979, −6.64750102636030109076526140515, −6.07612625835671437958698781781, −4.93829270955313866707210392416, −4.44167960679217612244655058665, −3.77691456065238663951292905625, −2.15957980678283354149844771387, −0.815938864956759250654776948299, 1.01655720113855350479401227133, 2.24138134743078567518116854280, 3.10523301201736614946997199841, 4.08114653613159477997623634677, 4.45522444175271612287460895602, 5.58388714063523793707826081874, 6.77899092865373241695935835514, 7.34098011540527379120250595679, 8.067311334129004883022851940923, 8.964139679776065585923844317789

Graph of the ZZ-function along the critical line