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Properties

Label 2-980-20.7-c1-0-22
Degree 22
Conductor 980980
Sign −0.820+0.572i-0.820 + 0.572i
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  + (0.0374 + 1.41i)2-s + (1.81 + 1.81i)3-s + (−1.99 + 0.105i)4-s + (0.00568 + 2.23i)5-s + (−2.49 + 2.63i)6-s + (−0.224 − 2.81i)8-s + 3.58i·9-s + (−3.16 + 0.0918i)10-s − 1.84i·11-s + (−3.81 − 3.43i)12-s + (−2.94 + 2.94i)13-s + (−4.04 + 4.06i)15-s + (3.97 − 0.423i)16-s + (2.17 + 2.17i)17-s + (−5.07 + 0.134i)18-s − 5.32·19-s + ⋯
L(s)  = 1  + (0.0264 + 0.999i)2-s + (1.04 + 1.04i)3-s + (−0.998 + 0.0529i)4-s + (0.00254 + 0.999i)5-s + (−1.01 + 1.07i)6-s + (−0.0793 − 0.996i)8-s + 1.19i·9-s + (−0.999 + 0.0290i)10-s − 0.555i·11-s + (−1.10 − 0.990i)12-s + (−0.817 + 0.817i)13-s + (−1.04 + 1.05i)15-s + (0.994 − 0.105i)16-s + (0.527 + 0.527i)17-s + (−1.19 + 0.0316i)18-s − 1.22·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.820+0.572i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.820+0.572i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.820+0.572i-0.820 + 0.572i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(687,⋅)\chi_{980} (687, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.820+0.572i)(2,\ 980,\ (\ :1/2),\ -0.820 + 0.572i)

Particular Values

L(1)L(1) ≈\approx 0.496608−1.57990i0.496608 - 1.57990i
L(12)L(\frac12) ≈\approx 0.496608−1.57990i0.496608 - 1.57990i
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+(−0.0374−1.41i)T 1 + (-0.0374 - 1.41i)T
5 1+(−0.00568−2.23i)T 1 + (-0.00568 - 2.23i)T
7 1 1
good3 1+(−1.81−1.81i)T+3iT2 1 + (-1.81 - 1.81i)T + 3iT^{2}
11 1+1.84iT−11T2 1 + 1.84iT - 11T^{2}
13 1+(2.94−2.94i)T−13iT2 1 + (2.94 - 2.94i)T - 13iT^{2}
17 1+(−2.17−2.17i)T+17iT2 1 + (-2.17 - 2.17i)T + 17iT^{2}
19 1+5.32T+19T2 1 + 5.32T + 19T^{2}
23 1+(1.78+1.78i)T+23iT2 1 + (1.78 + 1.78i)T + 23iT^{2}
29 1−6.35iT−29T2 1 - 6.35iT - 29T^{2}
31 1−4.36iT−31T2 1 - 4.36iT - 31T^{2}
37 1+(2.83+2.83i)T+37iT2 1 + (2.83 + 2.83i)T + 37iT^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1+(1.02+1.02i)T+43iT2 1 + (1.02 + 1.02i)T + 43iT^{2}
47 1+(−0.575+0.575i)T−47iT2 1 + (-0.575 + 0.575i)T - 47iT^{2}
53 1+(−1.91+1.91i)T−53iT2 1 + (-1.91 + 1.91i)T - 53iT^{2}
59 1−4.22T+59T2 1 - 4.22T + 59T^{2}
61 1−12.0T+61T2 1 - 12.0T + 61T^{2}
67 1+(−2.33+2.33i)T−67iT2 1 + (-2.33 + 2.33i)T - 67iT^{2}
71 1−10.5iT−71T2 1 - 10.5iT - 71T^{2}
73 1+(1.44−1.44i)T−73iT2 1 + (1.44 - 1.44i)T - 73iT^{2}
79 1−7.91T+79T2 1 - 7.91T + 79T^{2}
83 1+(0.227+0.227i)T+83iT2 1 + (0.227 + 0.227i)T + 83iT^{2}
89 1−4.33iT−89T2 1 - 4.33iT - 89T^{2}
97 1+(−0.196−0.196i)T+97iT2 1 + (-0.196 - 0.196i)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

−10.26719560565761311638203636183, −9.509173633578253771924101475935, −8.757059448296399674606023934878, −8.127456000976996508571427438732, −7.13418897905256960784185635297, −6.39683276684405933250015357582, −5.27759557744425971421281603736, −4.16271606537459724445544372331, −3.59154350019333902669988870313, −2.43953473967114103357766431439, 0.65980515141689291726300593345, 1.95150083537447275492871559011, 2.62828229851296085410554369720, 3.91500724524919642931833967974, 4.86253973006192713386032566173, 5.92638187272052374688738944652, 7.43254514559711203263500019244, 8.002047055798072080339912316859, 8.654757214720575452825255589366, 9.550427930653425930717539072373

Graph of the ZZ-function along the critical line