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Properties

Label 2-3332-476.291-c0-0-1
Degree 22
Conductor 33323332
Sign 0.233−0.972i0.233 - 0.972i
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 + (0.465 − 0.607i)5-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.465 + 0.607i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.499 − 0.866i)18-s + (−0.707 + 0.292i)20-s + (0.107 + 0.400i)25-s + (−1.93 − 0.517i)26-s + (0.707 + 1.70i)29-s + (−0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.465 − 0.607i)5-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.465 + 0.607i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.499 − 0.866i)18-s + (−0.707 + 0.292i)20-s + (0.107 + 0.400i)25-s + (−1.93 − 0.517i)26-s + (0.707 + 1.70i)29-s + (−0.965 + 0.258i)32-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.233−0.972i0.233 - 0.972i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1243,⋅)\chi_{3332} (1243, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.233−0.972i)(2,\ 3332,\ (\ :0),\ 0.233 - 0.972i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.98577152300.9857715230
L(12)L(\frac12) ≈\approx 0.98577152300.9857715230
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+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
good3 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
5 1+(−0.465+0.607i)T+(−0.258−0.965i)T2 1 + (-0.465 + 0.607i)T + (-0.258 - 0.965i)T^{2}
11 1+(−0.258+0.965i)T2 1 + (-0.258 + 0.965i)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+(0.965−0.258i)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+(−0.707+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+(−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+(−1.83+0.241i)T+(0.965−0.258i)T2 1 + (-1.83 + 0.241i)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+(0.758+0.0999i)T+(0.965+0.258i)T2 1 + (0.758 + 0.0999i)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+(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

−8.919338377945904144486023755770, −8.468561440739759946931927091702, −7.27810519443825709291573970181, −6.79613532611118150881407779943, −6.03571057831192426949083087946, −5.22001975818756266733312897242, −4.67123004351753622853373932093, −3.70711713474621701676871875243, −2.33937338753010755157741205633, −1.09655978153054501608470272356, 0.789264978740373632525103578207, 2.44780871946282110045119261734, 2.68506171603975328910482062844, 3.71650527517744901867133805097, 4.66698802627260816674083152845, 5.78745067076040962232664430262, 6.04419649244672509166064563605, 7.47043962141140739613812643872, 8.192471943427604980503822379266, 8.513778919643372687266564169706

Graph of the ZZ-function along the critical line