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Properties

Label 2-3332-476.247-c0-0-1
Degree 22
Conductor 33323332
Sign −0.972+0.233i-0.972 + 0.233i
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 + (1.12 + 1.46i)5-s + (−0.707 − 0.707i)8-s + (−0.965 + 0.258i)9-s + (−1.12 + 1.46i)10-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (−0.499 − 0.866i)18-s + (−1.70 − 0.707i)20-s + (−0.624 + 2.33i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s − i·34-s + (0.707 − 0.707i)36-s + (−1.46 + 1.12i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.12 + 1.46i)5-s + (−0.707 − 0.707i)8-s + (−0.965 + 0.258i)9-s + (−1.12 + 1.46i)10-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (−0.499 − 0.866i)18-s + (−1.70 − 0.707i)20-s + (−0.624 + 2.33i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s − i·34-s + (0.707 − 0.707i)36-s + (−1.46 + 1.12i)37-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.972+0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.972+0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.972+0.233i-0.972 + 0.233i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2627,⋅)\chi_{3332} (2627, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.972+0.233i)(2,\ 3332,\ (\ :0),\ -0.972 + 0.233i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1424403031.142440303
L(12)L(\frac12) ≈\approx 1.1424403031.142440303
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+(−1.12−1.46i)T+(−0.258+0.965i)T2 1 + (-1.12 - 1.46i)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+(0.707−1.70i)T+(−0.707−0.707i)T2 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2}
31 1+(0.965−0.258i)T2 1 + (0.965 - 0.258i)T^{2}
37 1+(1.46−1.12i)T+(0.258−0.965i)T2 1 + (1.46 - 1.12i)T + (0.258 - 0.965i)T^{2}
41 1+(0.292+0.707i)T+(−0.707+0.707i)T2 1 + (0.292 + 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+(−0.758−0.0999i)T+(0.965+0.258i)T2 1 + (-0.758 - 0.0999i)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+(−1.83+0.241i)T+(0.965−0.258i)T2 1 + (-1.83 + 0.241i)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+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)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

−8.968604085769015501170310569408, −8.550003453549906757520847215807, −7.36326093813315675667726618535, −6.90454733560109414127536949830, −6.29800337338599488824301582589, −5.53836384695720570934189535091, −5.01896021867004421173504562248, −3.62955722604908126287703879846, −2.96917741610359589758718964647, −2.04036288934781870613153549600, 0.58539248629331278947778383125, 1.87578948984808099513278947038, 2.43011682657594202268918753013, 3.74648874306236670675154747446, 4.50990933550106358646456071087, 5.41002486286836278034573531181, 5.71905655997961533965453186555, 6.60694446302236595815407365992, 8.189322803013431255270351322054, 8.605806865736493017659059497601

Graph of the ZZ-function along the critical line