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Properties

Label 2-980-35.13-c1-0-3
Degree 22
Conductor 980980
Sign −0.262−0.964i-0.262 - 0.964i
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  + (−1.42 + 1.42i)3-s + (−2.11 − 0.724i)5-s − 1.06i·9-s + 5.50·11-s + (1.36 − 1.36i)13-s + (4.04 − 1.98i)15-s + (0.849 + 0.849i)17-s − 0.519·19-s + (−4.91 − 4.91i)23-s + (3.95 + 3.06i)25-s + (−2.75 − 2.75i)27-s + 9.66i·29-s + 3.60i·31-s + (−7.85 + 7.85i)33-s + (−1.27 + 1.27i)37-s + ⋯
L(s)  = 1  + (−0.823 + 0.823i)3-s + (−0.946 − 0.323i)5-s − 0.354i·9-s + 1.66·11-s + (0.379 − 0.379i)13-s + (1.04 − 0.512i)15-s + (0.205 + 0.205i)17-s − 0.119·19-s + (−1.02 − 1.02i)23-s + (0.790 + 0.612i)25-s + (−0.530 − 0.530i)27-s + 1.79i·29-s + 0.646i·31-s + (−1.36 + 1.36i)33-s + (−0.210 + 0.210i)37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.262−0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.262−0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.262−0.964i-0.262 - 0.964i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(293,⋅)\chi_{980} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.262−0.964i)(2,\ 980,\ (\ :1/2),\ -0.262 - 0.964i)

Particular Values

L(1)L(1) ≈\approx 0.521354+0.682355i0.521354 + 0.682355i
L(12)L(\frac12) ≈\approx 0.521354+0.682355i0.521354 + 0.682355i
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+(2.11+0.724i)T 1 + (2.11 + 0.724i)T
7 1 1
good3 1+(1.42−1.42i)T−3iT2 1 + (1.42 - 1.42i)T - 3iT^{2}
11 1−5.50T+11T2 1 - 5.50T + 11T^{2}
13 1+(−1.36+1.36i)T−13iT2 1 + (-1.36 + 1.36i)T - 13iT^{2}
17 1+(−0.849−0.849i)T+17iT2 1 + (-0.849 - 0.849i)T + 17iT^{2}
19 1+0.519T+19T2 1 + 0.519T + 19T^{2}
23 1+(4.91+4.91i)T+23iT2 1 + (4.91 + 4.91i)T + 23iT^{2}
29 1−9.66iT−29T2 1 - 9.66iT - 29T^{2}
31 1−3.60iT−31T2 1 - 3.60iT - 31T^{2}
37 1+(1.27−1.27i)T−37iT2 1 + (1.27 - 1.27i)T - 37iT^{2}
41 1−9.63iT−41T2 1 - 9.63iT - 41T^{2}
43 1+(−5.88−5.88i)T+43iT2 1 + (-5.88 - 5.88i)T + 43iT^{2}
47 1+(−3.91−3.91i)T+47iT2 1 + (-3.91 - 3.91i)T + 47iT^{2}
53 1+(6.06+6.06i)T+53iT2 1 + (6.06 + 6.06i)T + 53iT^{2}
59 1+4.02T+59T2 1 + 4.02T + 59T^{2}
61 1−12.0iT−61T2 1 - 12.0iT - 61T^{2}
67 1+(10.1−10.1i)T−67iT2 1 + (10.1 - 10.1i)T - 67iT^{2}
71 1+1.96T+71T2 1 + 1.96T + 71T^{2}
73 1+(−10.1+10.1i)T−73iT2 1 + (-10.1 + 10.1i)T - 73iT^{2}
79 1+9.86iT−79T2 1 + 9.86iT - 79T^{2}
83 1+(5.05−5.05i)T−83iT2 1 + (5.05 - 5.05i)T - 83iT^{2}
89 1−4.50T+89T2 1 - 4.50T + 89T^{2}
97 1+(−10.3−10.3i)T+97iT2 1 + (-10.3 - 10.3i)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

−10.48369516588753548309411307405, −9.407745864410117187894919163847, −8.685358718480950727745139241087, −7.82233631642868217695237323006, −6.68614590770836310029261562760, −5.93061476970548194948418761684, −4.77639324748937834104178628185, −4.21159221821758405408268442199, −3.30942395806500690640253757322, −1.22219381293070847417423454872, 0.52426009799280054126164288562, 1.86457297625212403198618803460, 3.64760713269628767178560950998, 4.21803327029112316266952461930, 5.77169021248879438090418241572, 6.36856872757139123411020508530, 7.17143387064232417787262741502, 7.81124137888838073294448954871, 8.917976584625241327525823536572, 9.706228873984825366796679903621

Graph of the ZZ-function along the critical line