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Properties

Label 2-980-5.4-c1-0-2
Degree 22
Conductor 980980
Sign −0.632−0.774i-0.632 - 0.774i
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 + (−1.41 − 1.73i)5-s − 11-s + 1.73i·13-s + (2.99 − 2.44i)15-s + 5.19i·17-s − 2.82·19-s + 2.44i·23-s + (−0.999 + 4.89i)25-s + 5.19i·27-s + 7·29-s − 7.07·31-s − 1.73i·33-s + 7.34i·37-s − 2.99·39-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.632 − 0.774i)5-s − 0.301·11-s + 0.480i·13-s + (0.774 − 0.632i)15-s + 1.26i·17-s − 0.648·19-s + 0.510i·23-s + (−0.199 + 0.979i)25-s + 1.00i·27-s + 1.29·29-s − 1.27·31-s − 0.301i·33-s + 1.20i·37-s − 0.480·39-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.632−0.774i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.632−0.774i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.632−0.774i-0.632 - 0.774i
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), −0.632−0.774i)(2,\ 980,\ (\ :1/2),\ -0.632 - 0.774i)

Particular Values

L(1)L(1) ≈\approx 0.419536+0.884169i0.419536 + 0.884169i
L(12)L(\frac12) ≈\approx 0.419536+0.884169i0.419536 + 0.884169i
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+(1.41+1.73i)T 1 + (1.41 + 1.73i)T
7 1 1
good3 1−1.73iT−3T2 1 - 1.73iT - 3T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1−1.73iT−13T2 1 - 1.73iT - 13T^{2}
17 1−5.19iT−17T2 1 - 5.19iT - 17T^{2}
19 1+2.82T+19T2 1 + 2.82T + 19T^{2}
23 1−2.44iT−23T2 1 - 2.44iT - 23T^{2}
29 1−7T+29T2 1 - 7T + 29T^{2}
31 1+7.07T+31T2 1 + 7.07T + 31T^{2}
37 1−7.34iT−37T2 1 - 7.34iT - 37T^{2}
41 1+7.07T+41T2 1 + 7.07T + 41T^{2}
43 1+9.79iT−43T2 1 + 9.79iT - 43T^{2}
47 1−12.1iT−47T2 1 - 12.1iT - 47T^{2}
53 1−12.2iT−53T2 1 - 12.2iT - 53T^{2}
59 1+7.07T+59T2 1 + 7.07T + 59T^{2}
61 1−14.1T+61T2 1 - 14.1T + 61T^{2}
67 1+12.2iT−67T2 1 + 12.2iT - 67T^{2}
71 1+10T+71T2 1 + 10T + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+3T+79T2 1 + 3T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+5.19iT−97T2 1 + 5.19iT - 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

−10.36491174494227399064044545384, −9.418078231249830883339448985963, −8.732085221206297028357087132275, −8.015053077645827115910096070496, −6.97698557531034818077968814631, −5.81398345663655089806799938639, −4.79381452988828101998530296666, −4.21177954198269661181060492219, −3.34906453776074021685784432036, −1.58661233479883260522956626911, 0.45562131095881848179281836825, 2.14361889691587553951370284885, 3.09910867758932953131867298308, 4.30203119775406946387832885858, 5.46956454392757222239026486666, 6.71213006440912943594182825039, 7.02429709409911982384316956666, 7.913781972609144174578170141543, 8.568894742804875439753322715930, 9.866649520626739805001679251060

Graph of the ZZ-function along the critical line