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Properties

Label 2-980-35.3-c1-0-17
Degree 22
Conductor 980980
Sign 0.108+0.994i0.108 + 0.994i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.144 − 0.540i)3-s + (1.50 − 1.65i)5-s + (2.32 − 1.34i)9-s + (2.49 − 4.32i)11-s + (0.796 − 0.796i)13-s + (−1.11 − 0.575i)15-s + (−5.33 + 1.42i)17-s + (1.10 + 1.91i)19-s + (−1.89 + 7.08i)23-s + (−0.457 − 4.97i)25-s + (−2.25 − 2.25i)27-s − 2.10i·29-s + (−3.49 − 2.01i)31-s + (−2.70 − 0.723i)33-s + (9.23 + 2.47i)37-s + ⋯
L(s)  = 1  + (−0.0836 − 0.312i)3-s + (0.673 − 0.738i)5-s + (0.775 − 0.447i)9-s + (0.753 − 1.30i)11-s + (0.220 − 0.220i)13-s + (−0.286 − 0.148i)15-s + (−1.29 + 0.346i)17-s + (0.253 + 0.438i)19-s + (−0.395 + 1.47i)23-s + (−0.0914 − 0.995i)25-s + (−0.433 − 0.433i)27-s − 0.391i·29-s + (−0.627 − 0.362i)31-s + (−0.470 − 0.126i)33-s + (1.51 + 0.406i)37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.108+0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.108+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.108+0.994i0.108 + 0.994i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(913,⋅)\chi_{980} (913, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.108+0.994i)(2,\ 980,\ (\ :1/2),\ 0.108 + 0.994i)

Particular Values

L(1)L(1) ≈\approx 1.35086−1.21086i1.35086 - 1.21086i
L(12)L(\frac12) ≈\approx 1.35086−1.21086i1.35086 - 1.21086i
L(32)L(\frac{3}{2}) 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 1
5 1+(−1.50+1.65i)T 1 + (-1.50 + 1.65i)T
7 1 1
good3 1+(0.144+0.540i)T+(−2.59+1.5i)T2 1 + (0.144 + 0.540i)T + (-2.59 + 1.5i)T^{2}
11 1+(−2.49+4.32i)T+(−5.5−9.52i)T2 1 + (-2.49 + 4.32i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.796+0.796i)T−13iT2 1 + (-0.796 + 0.796i)T - 13iT^{2}
17 1+(5.33−1.42i)T+(14.7−8.5i)T2 1 + (5.33 - 1.42i)T + (14.7 - 8.5i)T^{2}
19 1+(−1.10−1.91i)T+(−9.5+16.4i)T2 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.89−7.08i)T+(−19.9−11.5i)T2 1 + (1.89 - 7.08i)T + (-19.9 - 11.5i)T^{2}
29 1+2.10iT−29T2 1 + 2.10iT - 29T^{2}
31 1+(3.49+2.01i)T+(15.5+26.8i)T2 1 + (3.49 + 2.01i)T + (15.5 + 26.8i)T^{2}
37 1+(−9.23−2.47i)T+(32.0+18.5i)T2 1 + (-9.23 - 2.47i)T + (32.0 + 18.5i)T^{2}
41 1+9.32iT−41T2 1 + 9.32iT - 41T^{2}
43 1+(−3.09−3.09i)T+43iT2 1 + (-3.09 - 3.09i)T + 43iT^{2}
47 1+(−0.781+2.91i)T+(−40.7−23.5i)T2 1 + (-0.781 + 2.91i)T + (-40.7 - 23.5i)T^{2}
53 1+(−3.48+0.933i)T+(45.8−26.5i)T2 1 + (-3.48 + 0.933i)T + (45.8 - 26.5i)T^{2}
59 1+(4.73−8.20i)T+(−29.5−51.0i)T2 1 + (4.73 - 8.20i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.50+0.866i)T+(30.5−52.8i)T2 1 + (-1.50 + 0.866i)T + (30.5 - 52.8i)T^{2}
67 1+(2.73+10.1i)T+(−58.0+33.5i)T2 1 + (2.73 + 10.1i)T + (-58.0 + 33.5i)T^{2}
71 1+2.88T+71T2 1 + 2.88T + 71T^{2}
73 1+(2.79+10.4i)T+(−63.2+36.5i)T2 1 + (2.79 + 10.4i)T + (-63.2 + 36.5i)T^{2}
79 1+(3.32−1.91i)T+(39.5−68.4i)T2 1 + (3.32 - 1.91i)T + (39.5 - 68.4i)T^{2}
83 1+(11.9−11.9i)T−83iT2 1 + (11.9 - 11.9i)T - 83iT^{2}
89 1+(1.81+3.15i)T+(−44.5+77.0i)T2 1 + (1.81 + 3.15i)T + (-44.5 + 77.0i)T^{2}
97 1+(−8.67−8.67i)T+97iT2 1 + (-8.67 - 8.67i)T + 97iT^{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

−9.581188153705297954050245870721, −9.092787816877817358699422543497, −8.246687014480377315124998274204, −7.25407790541464848162057670745, −6.10776589796053660413591660683, −5.82478901261521123115282950471, −4.41971225133709541760193442632, −3.58283882109084952619541388698, −1.95929183264896194414224136242, −0.905852862284589040585356639319, 1.72428702340613975687276322266, 2.65165947067439721655850196429, 4.21047990957067530061018128474, 4.70958088313362184885430137166, 6.05648808633705936115186659000, 6.88684865566464400658132210054, 7.35644410292126293269792294976, 8.764820793302869856270494372242, 9.543595209260014068372909336037, 10.10512525553457852578607546828

Graph of the ZZ-function along the critical line