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Properties

Label 2-980-140.139-c1-0-1
Degree 22
Conductor 980980
Sign 0.987−0.155i0.987 - 0.155i
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  + (−0.255 − 1.39i)2-s − 3.18i·3-s + (−1.86 + 0.711i)4-s + (−1.80 − 1.31i)5-s + (−4.43 + 0.815i)6-s + (1.46 + 2.41i)8-s − 7.15·9-s + (−1.36 + 2.85i)10-s + 4.51i·11-s + (2.26 + 5.95i)12-s + 2.22·13-s + (−4.19 + 5.75i)15-s + (2.98 − 2.66i)16-s + 2.52·17-s + (1.83 + 9.94i)18-s − 5.21·19-s + ⋯
L(s)  = 1  + (−0.180 − 0.983i)2-s − 1.83i·3-s + (−0.934 + 0.355i)4-s + (−0.808 − 0.588i)5-s + (−1.80 + 0.332i)6-s + (0.519 + 0.854i)8-s − 2.38·9-s + (−0.432 + 0.901i)10-s + 1.36i·11-s + (0.654 + 1.71i)12-s + 0.617·13-s + (−1.08 + 1.48i)15-s + (0.746 − 0.665i)16-s + 0.613·17-s + (0.431 + 2.34i)18-s − 1.19·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.987−0.155i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.987−0.155i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.987−0.155i0.987 - 0.155i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(979,⋅)\chi_{980} (979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.987−0.155i)(2,\ 980,\ (\ :1/2),\ 0.987 - 0.155i)

Particular Values

L(1)L(1) ≈\approx 0.0585708+0.00456803i0.0585708 + 0.00456803i
L(12)L(\frac12) ≈\approx 0.0585708+0.00456803i0.0585708 + 0.00456803i
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+(0.255+1.39i)T 1 + (0.255 + 1.39i)T
5 1+(1.80+1.31i)T 1 + (1.80 + 1.31i)T
7 1 1
good3 1+3.18iT−3T2 1 + 3.18iT - 3T^{2}
11 1−4.51iT−11T2 1 - 4.51iT - 11T^{2}
13 1−2.22T+13T2 1 - 2.22T + 13T^{2}
17 1−2.52T+17T2 1 - 2.52T + 17T^{2}
19 1+5.21T+19T2 1 + 5.21T + 19T^{2}
23 1+1.71T+23T2 1 + 1.71T + 23T^{2}
29 1+2.31T+29T2 1 + 2.31T + 29T^{2}
31 1+4.62T+31T2 1 + 4.62T + 31T^{2}
37 1+0.336iT−37T2 1 + 0.336iT - 37T^{2}
41 1−3.28iT−41T2 1 - 3.28iT - 41T^{2}
43 1+6.66T+43T2 1 + 6.66T + 43T^{2}
47 1+1.44iT−47T2 1 + 1.44iT - 47T^{2}
53 1+10.0iT−53T2 1 + 10.0iT - 53T^{2}
59 1+3.20T+59T2 1 + 3.20T + 59T^{2}
61 1−6.05iT−61T2 1 - 6.05iT - 61T^{2}
67 1+11.1T+67T2 1 + 11.1T + 67T^{2}
71 1−9.15iT−71T2 1 - 9.15iT - 71T^{2}
73 1+3.24T+73T2 1 + 3.24T + 73T^{2}
79 1−14.2iT−79T2 1 - 14.2iT - 79T^{2}
83 1+11.3iT−83T2 1 + 11.3iT - 83T^{2}
89 1+15.2iT−89T2 1 + 15.2iT - 89T^{2}
97 1−4.49T+97T2 1 - 4.49T + 97T^{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

−10.10248197086586251374156161763, −8.948634328968352387629504835286, −8.333375885210607635429211416127, −7.64916505608511568100547127166, −6.95337969062812357787386524227, −5.72318156809775550970109897559, −4.58303605249822891869279623967, −3.46265081698509650813120038202, −2.12803543675313195275432005969, −1.36240626114489518638863736639, 0.03073425565956456117160118871, 3.26757429355933890290551772052, 3.81320717641160889489839966853, 4.65940329975623690257479929180, 5.72746910253823979590998429142, 6.30130813880881510649469949723, 7.67876197769553985473105604012, 8.506593809172169452580460588727, 8.944409427162867363776627477989, 9.940154763373168229873236718027

Graph of the ZZ-function along the critical line