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Properties

Label 2-980-140.3-c0-0-0
Degree 22
Conductor 980980
Sign −0.442−0.896i-0.442 - 0.896i
Analytic cond. 0.4890830.489083
Root an. cond. 0.6993450.699345
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 + (−0.793 − 0.608i)5-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.608 + 0.793i)10-s + (−1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)18-s + (−0.382 − 0.923i)20-s + (0.258 + 0.965i)25-s + (1.60 − 0.923i)26-s + 1.41i·29-s + (−0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.793 − 0.608i)5-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.608 + 0.793i)10-s + (−1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)18-s + (−0.382 − 0.923i)20-s + (0.258 + 0.965i)25-s + (1.60 − 0.923i)26-s + 1.41i·29-s + (−0.258 − 0.965i)32-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.442−0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s) L(s)=((−0.442−0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.442−0.896i-0.442 - 0.896i
Analytic conductor: 0.4890830.489083
Root analytic conductor: 0.6993450.699345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ980(423,⋅)\chi_{980} (423, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :0), −0.442−0.896i)(2,\ 980,\ (\ :0),\ -0.442 - 0.896i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20986980850.2098698085
L(12)L(\frac12) ≈\approx 0.20986980850.2098698085
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
5 1+(0.793+0.608i)T 1 + (0.793 + 0.608i)T
7 1 1
good3 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(1.30−1.30i)T−iT2 1 + (1.30 - 1.30i)T - iT^{2}
17 1+(0.739−0.198i)T+(0.866−0.5i)T2 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
41 1+0.765iT−T2 1 + 0.765iT - T^{2}
43 1−iT2 1 - iT^{2}
47 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(1.60−0.923i)T+(0.5−0.866i)T2 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2}
67 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.198+0.739i)T+(−0.866+0.5i)T2 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(−0.923−1.60i)T+(−0.5+0.866i)T2 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2}
97 1+(−1.30−1.30i)T+iT2 1 + (-1.30 - 1.30i)T + iT^{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

−10.58509331467899059631913064337, −9.273059632179850819792288147242, −9.007749242780114431572499046949, −8.071453204772221924978167558190, −7.36070479481897925839917517709, −6.54970014253360384761863622464, −5.18747970343789059485017474280, −4.22979738489037623592835332475, −2.94868476808694279047656894752, −1.80173767710441374026707575090, 0.25224155225769042694524968774, 2.47895911302942702865657782955, 3.21953592942144381038144394182, 4.77912142873799789323165718452, 5.93274768787932598122853440615, 6.69051636225399937448884141736, 7.70118326770125708436697465717, 8.056571670827168070611320271409, 9.093945163117925026122343967201, 9.903195382520883118680170589129

Graph of the ZZ-function along the critical line