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Properties

Label 2-980-28.27-c1-0-9
Degree 22
Conductor 980980
Sign −0.0773−0.997i-0.0773 - 0.997i
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  + (−1.34 − 0.442i)2-s − 0.901·3-s + (1.60 + 1.18i)4-s + i·5-s + (1.21 + 0.398i)6-s + (−1.63 − 2.30i)8-s − 2.18·9-s + (0.442 − 1.34i)10-s − 3.74i·11-s + (−1.44 − 1.07i)12-s + 2.41i·13-s − 0.901i·15-s + (1.17 + 3.82i)16-s − 0.583i·17-s + (2.93 + 0.967i)18-s + 6.15·19-s + ⋯
L(s)  = 1  + (−0.949 − 0.312i)2-s − 0.520·3-s + (0.804 + 0.594i)4-s + 0.447i·5-s + (0.494 + 0.162i)6-s + (−0.578 − 0.815i)8-s − 0.729·9-s + (0.139 − 0.424i)10-s − 1.12i·11-s + (−0.418 − 0.309i)12-s + 0.671i·13-s − 0.232i·15-s + (0.293 + 0.955i)16-s − 0.141i·17-s + (0.692 + 0.228i)18-s + 1.41·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.0773−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0773 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.0773−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0773 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.0773−0.997i-0.0773 - 0.997i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(391,⋅)\chi_{980} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.0773−0.997i)(2,\ 980,\ (\ :1/2),\ -0.0773 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.327187+0.353565i0.327187 + 0.353565i
L(12)L(\frac12) ≈\approx 0.327187+0.353565i0.327187 + 0.353565i
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+(1.34+0.442i)T 1 + (1.34 + 0.442i)T
5 1−iT 1 - iT
7 1 1
good3 1+0.901T+3T2 1 + 0.901T + 3T^{2}
11 1+3.74iT−11T2 1 + 3.74iT - 11T^{2}
13 1−2.41iT−13T2 1 - 2.41iT - 13T^{2}
17 1+0.583iT−17T2 1 + 0.583iT - 17T^{2}
19 1−6.15T+19T2 1 - 6.15T + 19T^{2}
23 1−4.31iT−23T2 1 - 4.31iT - 23T^{2}
29 1+0.435T+29T2 1 + 0.435T + 29T^{2}
31 1−2.53T+31T2 1 - 2.53T + 31T^{2}
37 1+11.3T+37T2 1 + 11.3T + 37T^{2}
41 1−7.35iT−41T2 1 - 7.35iT - 41T^{2}
43 1−5.80iT−43T2 1 - 5.80iT - 43T^{2}
47 1+11.5T+47T2 1 + 11.5T + 47T^{2}
53 1+3.11T+53T2 1 + 3.11T + 53T^{2}
59 1−3.47T+59T2 1 - 3.47T + 59T^{2}
61 1−10.3iT−61T2 1 - 10.3iT - 61T^{2}
67 1−9.84iT−67T2 1 - 9.84iT - 67T^{2}
71 1+9.96iT−71T2 1 + 9.96iT - 71T^{2}
73 1−9.79iT−73T2 1 - 9.79iT - 73T^{2}
79 1−0.459iT−79T2 1 - 0.459iT - 79T^{2}
83 1+2.59T+83T2 1 + 2.59T + 83T^{2}
89 1−9.88iT−89T2 1 - 9.88iT - 89T^{2}
97 1−4.54iT−97T2 1 - 4.54iT - 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

−10.20768037284709525823450994557, −9.443476050342028976105520547215, −8.622660064359537180054782148369, −7.85012335446375397211927605967, −6.88132771167523501727048006105, −6.13665570921498613856783777899, −5.22036263799560420434979944769, −3.53785760250028641231909624145, −2.83453274847513984901259634389, −1.25187849366145909565766760715, 0.35141758459469523762374816199, 1.83437628404043453179665608182, 3.18299540381328235639311594359, 4.93396687169036100051406797646, 5.47234094758701315865337870288, 6.50954892427541359935705242363, 7.31747629765421286380959704523, 8.198549754217040371421530291092, 8.896791225522249357400337923944, 9.817556179519680284953946207419

Graph of the ZZ-function along the critical line