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Properties

Label 2-980-140.139-c1-0-80
Degree 22
Conductor 980980
Sign −0.0335+0.999i-0.0335 + 0.999i
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  + (−1.36 − 0.385i)2-s − 0.423i·3-s + (1.70 + 1.04i)4-s + (1.74 − 1.39i)5-s + (−0.163 + 0.576i)6-s + (−1.91 − 2.08i)8-s + 2.82·9-s + (−2.91 + 1.23i)10-s − 4.89i·11-s + (0.443 − 0.721i)12-s + 2.54·13-s + (−0.591 − 0.738i)15-s + (1.80 + 3.57i)16-s − 5.11·17-s + (−3.83 − 1.08i)18-s + 6.26·19-s + ⋯
L(s)  = 1  + (−0.962 − 0.272i)2-s − 0.244i·3-s + (0.851 + 0.524i)4-s + (0.780 − 0.625i)5-s + (−0.0665 + 0.235i)6-s + (−0.676 − 0.736i)8-s + 0.940·9-s + (−0.921 + 0.388i)10-s − 1.47i·11-s + (0.128 − 0.208i)12-s + 0.706·13-s + (−0.152 − 0.190i)15-s + (0.450 + 0.892i)16-s − 1.24·17-s + (−0.904 − 0.256i)18-s + 1.43·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.0335+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0335 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.0335+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0335 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.0335+0.999i-0.0335 + 0.999i
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), −0.0335+0.999i)(2,\ 980,\ (\ :1/2),\ -0.0335 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.873939−0.903739i0.873939 - 0.903739i
L(12)L(\frac12) ≈\approx 0.873939−0.903739i0.873939 - 0.903739i
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+(1.36+0.385i)T 1 + (1.36 + 0.385i)T
5 1+(−1.74+1.39i)T 1 + (-1.74 + 1.39i)T
7 1 1
good3 1+0.423iT−3T2 1 + 0.423iT - 3T^{2}
11 1+4.89iT−11T2 1 + 4.89iT - 11T^{2}
13 1−2.54T+13T2 1 - 2.54T + 13T^{2}
17 1+5.11T+17T2 1 + 5.11T + 17T^{2}
19 1−6.26T+19T2 1 - 6.26T + 19T^{2}
23 1+4.63T+23T2 1 + 4.63T + 23T^{2}
29 1+1.88T+29T2 1 + 1.88T + 29T^{2}
31 1+1.47T+31T2 1 + 1.47T + 31T^{2}
37 1+2.35iT−37T2 1 + 2.35iT - 37T^{2}
41 1−7.05iT−41T2 1 - 7.05iT - 41T^{2}
43 1−10.7T+43T2 1 - 10.7T + 43T^{2}
47 1−12.2iT−47T2 1 - 12.2iT - 47T^{2}
53 1+2.23iT−53T2 1 + 2.23iT - 53T^{2}
59 1+5.68T+59T2 1 + 5.68T + 59T^{2}
61 1+3.87iT−61T2 1 + 3.87iT - 61T^{2}
67 1+0.889T+67T2 1 + 0.889T + 67T^{2}
71 1+14.3iT−71T2 1 + 14.3iT - 71T^{2}
73 1+7.87T+73T2 1 + 7.87T + 73T^{2}
79 1+4.63iT−79T2 1 + 4.63iT - 79T^{2}
83 1+4.32iT−83T2 1 + 4.32iT - 83T^{2}
89 1−2.18iT−89T2 1 - 2.18iT - 89T^{2}
97 1−3.42T+97T2 1 - 3.42T + 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

−9.528805821842074310998471900524, −9.151473384117175381132148060550, −8.241057642095467474362910856091, −7.50715412446562702709687554763, −6.33678985323859092000106994561, −5.84859617521673763669525145826, −4.34978878799297992224456288488, −3.12892807189809310164980063130, −1.82564796081963552995737426588, −0.853044336688105131540031543319, 1.55115276311505078982875549075, 2.40694537702422390976874698185, 3.94653126336625394791384742158, 5.20785998500267038651613858071, 6.16734502263099378833724428411, 7.12984292618136733836354877023, 7.40208583894422020001389340684, 8.788963126707178725299088630041, 9.525720080845937124524649598245, 10.05184807462169006539803573940

Graph of the ZZ-function along the critical line