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Properties

Label 2-980-5.4-c1-0-9
Degree 22
Conductor 980980
Sign 0.955+0.293i0.955 + 0.293i
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  − 1.73i·3-s + (2.13 + 0.656i)5-s + 2.27·11-s + 6.09i·13-s + (1.13 − 3.70i)15-s + 4.77i·17-s + 4.27·19-s − 0.894i·23-s + (4.13 + 2.80i)25-s − 5.19i·27-s + 3.27·29-s − 4.27·31-s − 3.94i·33-s − 5.61i·37-s + 10.5·39-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.955 + 0.293i)5-s + 0.685·11-s + 1.68i·13-s + (0.293 − 0.955i)15-s + 1.15i·17-s + 0.980·19-s − 0.186i·23-s + (0.827 + 0.561i)25-s − 1.00i·27-s + 0.608·29-s − 0.767·31-s − 0.685i·33-s − 0.923i·37-s + 1.68·39-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.955+0.293i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.955+0.293i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.955+0.293i0.955 + 0.293i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(589,⋅)\chi_{980} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.955+0.293i)(2,\ 980,\ (\ :1/2),\ 0.955 + 0.293i)

Particular Values

L(1)L(1) ≈\approx 2.04021−0.306350i2.04021 - 0.306350i
L(12)L(\frac12) ≈\approx 2.04021−0.306350i2.04021 - 0.306350i
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+(−2.13−0.656i)T 1 + (-2.13 - 0.656i)T
7 1 1
good3 1+1.73iT−3T2 1 + 1.73iT - 3T^{2}
11 1−2.27T+11T2 1 - 2.27T + 11T^{2}
13 1−6.09iT−13T2 1 - 6.09iT - 13T^{2}
17 1−4.77iT−17T2 1 - 4.77iT - 17T^{2}
19 1−4.27T+19T2 1 - 4.27T + 19T^{2}
23 1+0.894iT−23T2 1 + 0.894iT - 23T^{2}
29 1−3.27T+29T2 1 - 3.27T + 29T^{2}
31 1+4.27T+31T2 1 + 4.27T + 31T^{2}
37 1+5.61iT−37T2 1 + 5.61iT - 37T^{2}
41 1+11.2T+41T2 1 + 11.2T + 41T^{2}
43 1−6.50iT−43T2 1 - 6.50iT - 43T^{2}
47 1−2.15iT−47T2 1 - 2.15iT - 47T^{2}
53 1+7.40iT−53T2 1 + 7.40iT - 53T^{2}
59 1+4.27T+59T2 1 + 4.27T + 59T^{2}
61 1−1.54T+61T2 1 - 1.54T + 61T^{2}
67 1+13.9iT−67T2 1 + 13.9iT - 67T^{2}
71 1−10.5T+71T2 1 - 10.5T + 71T^{2}
73 1+2.15iT−73T2 1 + 2.15iT - 73T^{2}
79 1−0.274T+79T2 1 - 0.274T + 79T^{2}
83 1+5.67iT−83T2 1 + 5.67iT - 83T^{2}
89 1−7T+89T2 1 - 7T + 89T^{2}
97 1+6.92iT−97T2 1 + 6.92iT - 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

−9.800054118687206859668603999647, −9.223899700719123256339317963550, −8.270127384898417845766557641650, −7.16053451874479466730082982188, −6.61569728695963710160373894604, −6.00934625557292535539049770195, −4.74155372741330535889366609832, −3.54953427983868203590556281641, −2.03410528293657867251889262524, −1.46707223415394666313660463887, 1.15338412196087249873991553514, 2.80284053534712485165337929509, 3.72775695241190780099359122473, 5.12262614404867285132776932416, 5.27761544728862968081407774533, 6.55189533247964646349836358030, 7.53312433866475062999696157685, 8.679168755041749053163975472392, 9.376549480145638012281000144870, 10.06034024267647370788569403097

Graph of the ZZ-function along the critical line