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Properties

Label 2-980-140.139-c1-0-21
Degree 22
Conductor 980980
Sign −0.975+0.219i-0.975 + 0.219i
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  + (1.23 + 0.691i)2-s + 2.08i·3-s + (1.04 + 1.70i)4-s + (−1.52 + 1.63i)5-s + (−1.43 + 2.56i)6-s + (0.107 + 2.82i)8-s − 1.32·9-s + (−3.01 + 0.967i)10-s − 0.775i·11-s + (−3.54 + 2.17i)12-s − 4.18·13-s + (−3.40 − 3.16i)15-s + (−1.82 + 3.56i)16-s + 4.18·17-s + (−1.63 − 0.917i)18-s − 4.88·19-s + ⋯
L(s)  = 1  + (0.872 + 0.488i)2-s + 1.20i·3-s + (0.521 + 0.853i)4-s + (−0.680 + 0.732i)5-s + (−0.587 + 1.04i)6-s + (0.0381 + 0.999i)8-s − 0.442·9-s + (−0.952 + 0.305i)10-s − 0.233i·11-s + (−1.02 + 0.626i)12-s − 1.16·13-s + (−0.879 − 0.817i)15-s + (−0.455 + 0.890i)16-s + 1.01·17-s + (−0.385 − 0.216i)18-s − 1.12·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.975+0.219i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.975+0.219i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.975+0.219i-0.975 + 0.219i
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), −0.975+0.219i)(2,\ 980,\ (\ :1/2),\ -0.975 + 0.219i)

Particular Values

L(1)L(1) ≈\approx 0.224642−2.02030i0.224642 - 2.02030i
L(12)L(\frac12) ≈\approx 0.224642−2.02030i0.224642 - 2.02030i
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+(−1.23−0.691i)T 1 + (-1.23 - 0.691i)T
5 1+(1.52−1.63i)T 1 + (1.52 - 1.63i)T
7 1 1
good3 1−2.08iT−3T2 1 - 2.08iT - 3T^{2}
11 1+0.775iT−11T2 1 + 0.775iT - 11T^{2}
13 1+4.18T+13T2 1 + 4.18T + 13T^{2}
17 1−4.18T+17T2 1 - 4.18T + 17T^{2}
19 1+4.88T+19T2 1 + 4.88T + 19T^{2}
23 1+1.89T+23T2 1 + 1.89T + 23T^{2}
29 1−9.98T+29T2 1 - 9.98T + 29T^{2}
31 1−10.3T+31T2 1 - 10.3T + 31T^{2}
37 1−3.41iT−37T2 1 - 3.41iT - 37T^{2}
41 1+3.02iT−41T2 1 + 3.02iT - 41T^{2}
43 1+9.19T+43T2 1 + 9.19T + 43T^{2}
47 1−8.27iT−47T2 1 - 8.27iT - 47T^{2}
53 1+2.59iT−53T2 1 + 2.59iT - 53T^{2}
59 1+4.60T+59T2 1 + 4.60T + 59T^{2}
61 1+3.26iT−61T2 1 + 3.26iT - 61T^{2}
67 1−2.27T+67T2 1 - 2.27T + 67T^{2}
71 1−4.41iT−71T2 1 - 4.41iT - 71T^{2}
73 1+2.74T+73T2 1 + 2.74T + 73T^{2}
79 1−14.2iT−79T2 1 - 14.2iT - 79T^{2}
83 1−2.36iT−83T2 1 - 2.36iT - 83T^{2}
89 1+14.3iT−89T2 1 + 14.3iT - 89T^{2}
97 1−14.4T+97T2 1 - 14.4T + 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

−10.30038843557478439155211430382, −9.995672828414270049367875090733, −8.530570224108336820922302801979, −7.916593798082230482525210977467, −6.90280347706216075378319128728, −6.15688825309408697973584362342, −4.86498674334837774986022429252, −4.44410968702183548245721173367, −3.41909192880302883732358977356, −2.66395270350521842216124773107, 0.69812420302899534066541484140, 1.87826293477550644549074371654, 2.97903116344241567480798116128, 4.34218663474838015895913612827, 4.95457227229005269052128808644, 6.16712295632553464960618544456, 6.91224727953287733018125634503, 7.75774435036875438966295513437, 8.477248390693047262927958196585, 9.825947903972653283646737655898

Graph of the ZZ-function along the critical line