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Properties

Label 2-980-7.4-c1-0-12
Degree 22
Conductor 980980
Sign −0.701+0.712i-0.701 + 0.712i
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  + (1.5 − 2.59i)3-s + (−0.5 − 0.866i)5-s + (−3 − 5.19i)9-s + (2.5 − 4.33i)11-s + 3·13-s − 3·15-s + (−0.5 + 0.866i)17-s + (3 + 5.19i)19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s − 9·27-s − 9·29-s + (−2 + 3.46i)31-s + (−7.50 − 12.9i)33-s + (−1 − 1.73i)37-s + ⋯
L(s)  = 1  + (0.866 − 1.49i)3-s + (−0.223 − 0.387i)5-s + (−1 − 1.73i)9-s + (0.753 − 1.30i)11-s + 0.832·13-s − 0.774·15-s + (−0.121 + 0.210i)17-s + (0.688 + 1.19i)19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s − 1.73·27-s − 1.67·29-s + (−0.359 + 0.622i)31-s + (−1.30 − 2.26i)33-s + (−0.164 − 0.284i)37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.701+0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.701+0.712i-0.701 + 0.712i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(361,⋅)\chi_{980} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.701+0.712i)(2,\ 980,\ (\ :1/2),\ -0.701 + 0.712i)

Particular Values

L(1)L(1) ≈\approx 0.786133−1.87584i0.786133 - 1.87584i
L(12)L(\frac12) ≈\approx 0.786133−1.87584i0.786133 - 1.87584i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1 1
good3 1+(−1.5+2.59i)T+(−1.5−2.59i)T2 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2}
11 1+(−2.5+4.33i)T+(−5.5−9.52i)T2 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2}
13 1−3T+13T2 1 - 3T + 13T^{2}
17 1+(0.5−0.866i)T+(−8.5−14.7i)T2 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3−5.19i)T+(−9.5+16.4i)T2 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+9T+29T2 1 + 9T + 29T^{2}
31 1+(2−3.46i)T+(−15.5−26.8i)T2 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2}
37 1+(1+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1−4T+41T2 1 - 4T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1+(0.5+0.866i)T+(−23.5+40.7i)T2 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2}
53 1+(2−3.46i)T+(−26.5−45.8i)T2 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(4−6.92i)T+(−29.5−51.0i)T2 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2}
61 1+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}
67 1+(6−10.3i)T+(−33.5−58.0i)T2 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+(−1+1.73i)T+(−36.5−63.2i)T2 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(6.5+11.2i)T+(−39.5+68.4i)T2 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1+(−2−3.46i)T+(−44.5+77.0i)T2 1 + (-2 - 3.46i)T + (-44.5 + 77.0i)T^{2}
97 1−13T+97T2 1 - 13T + 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

−9.170036079240335566035548870042, −8.780991723512763655898270630825, −7.984498717222100246955147740029, −7.39754411231380091219373760824, −6.22680407899718166068467139003, −5.82795065963223808062270303981, −3.95417231387729679149553435415, −3.23561328476068011249769120887, −1.86184011198517421290437146900, −0.893051422165929279927283314065, 2.06012136757807741944124969827, 3.29885871641974753156334332248, 3.99431508679137193633347955316, 4.75357227039929445424942766486, 5.85384386671571582830167338039, 7.18652279334543349240355008413, 7.84282940941575329803294748095, 9.144391342023484309781584764377, 9.314339183736443234315971302491, 10.07874805288484003330024534610

Graph of the ZZ-function along the critical line