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Properties

Label 2-3332-476.359-c0-0-0
Degree 22
Conductor 33323332
Sign −0.142−0.989i-0.142 - 0.989i
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 + (−1.83 − 0.241i)5-s + (−0.707 + 0.707i)8-s + (0.258 + 0.965i)9-s + (1.83 − 0.241i)10-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.499 − 0.866i)18-s + (−1.70 + 0.707i)20-s + (2.33 + 0.624i)25-s + (−0.707 − 1.70i)29-s + (−0.258 + 0.965i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−0.241 + 1.83i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.83 − 0.241i)5-s + (−0.707 + 0.707i)8-s + (0.258 + 0.965i)9-s + (1.83 − 0.241i)10-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.499 − 0.866i)18-s + (−1.70 + 0.707i)20-s + (2.33 + 0.624i)25-s + (−0.707 − 1.70i)29-s + (−0.258 + 0.965i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−0.241 + 1.83i)37-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.142−0.989i-0.142 - 0.989i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(3215,⋅)\chi_{3332} (3215, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.142−0.989i)(2,\ 3332,\ (\ :0),\ -0.142 - 0.989i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.37801769810.3780176981
L(12)L(\frac12) ≈\approx 0.37801769810.3780176981
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+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
good3 1+(−0.258−0.965i)T2 1 + (-0.258 - 0.965i)T^{2}
5 1+(1.83+0.241i)T+(0.965+0.258i)T2 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2}
11 1+(0.965−0.258i)T2 1 + (0.965 - 0.258i)T^{2}
13 1−T2 1 - T^{2}
19 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
23 1+(−0.258+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+(0.292−0.707i)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+(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+(0.465+0.607i)T+(−0.258+0.965i)T2 1 + (0.465 + 0.607i)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+(1.12−1.46i)T+(−0.258−0.965i)T2 1 + (1.12 - 1.46i)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+(−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

−8.820326321199279934147053790763, −8.017007572449845541935661207879, −7.77028542320002903380815931567, −7.16674662065826267922854708039, −6.27069439053920585203027812680, −5.09621884885345239523807194106, −4.51682642026728674104845192389, −3.41771687755488308128885370052, −2.46550938726616524475874508777, −1.05328748996031345716370664575, 0.38077843168179938311252986372, 1.72835739662762071265245408365, 3.30949861999462570812819989718, 3.53494915371445483046303017657, 4.42053803996716639669785761405, 5.80542508151205201063954352372, 6.78069591179218972629712007319, 7.30860066372174603237912866283, 7.82761073893858858827806033756, 8.807915988141060156976729614634

Graph of the ZZ-function along the critical line