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Properties

Label 2-980-5.4-c1-0-19
Degree 22
Conductor 980980
Sign −0.732−0.680i-0.732 - 0.680i
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 + (−1.63 − 1.52i)5-s − 5.27·11-s − 2.62i·13-s + (−2.63 + 2.83i)15-s + 0.418i·17-s − 3.27·19-s + 7.82i·23-s + (0.362 + 4.98i)25-s − 5.19i·27-s − 4.27·29-s + 3.27·31-s + 9.13i·33-s − 9.97i·37-s − 4.54·39-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.732 − 0.680i)5-s − 1.59·11-s − 0.728i·13-s + (−0.680 + 0.732i)15-s + 0.101i·17-s − 0.751·19-s + 1.63i·23-s + (0.0725 + 0.997i)25-s − 1.00i·27-s − 0.793·29-s + 0.588·31-s + 1.59i·33-s − 1.63i·37-s − 0.728·39-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.732−0.680i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.732−0.680i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.732−0.680i-0.732 - 0.680i
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), −0.732−0.680i)(2,\ 980,\ (\ :1/2),\ -0.732 - 0.680i)

Particular Values

L(1)L(1) ≈\approx 0.122424+0.311423i0.122424 + 0.311423i
L(12)L(\frac12) ≈\approx 0.122424+0.311423i0.122424 + 0.311423i
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+(1.63+1.52i)T 1 + (1.63 + 1.52i)T
7 1 1
good3 1+1.73iT−3T2 1 + 1.73iT - 3T^{2}
11 1+5.27T+11T2 1 + 5.27T + 11T^{2}
13 1+2.62iT−13T2 1 + 2.62iT - 13T^{2}
17 1−0.418iT−17T2 1 - 0.418iT - 17T^{2}
19 1+3.27T+19T2 1 + 3.27T + 19T^{2}
23 1−7.82iT−23T2 1 - 7.82iT - 23T^{2}
29 1+4.27T+29T2 1 + 4.27T + 29T^{2}
31 1−3.27T+31T2 1 - 3.27T + 31T^{2}
37 1+9.97iT−37T2 1 + 9.97iT - 37T^{2}
41 1+3.72T+41T2 1 + 3.72T + 41T^{2}
43 1−2.15iT−43T2 1 - 2.15iT - 43T^{2}
47 1−6.50iT−47T2 1 - 6.50iT - 47T^{2}
53 1−5.67iT−53T2 1 - 5.67iT - 53T^{2}
59 1−3.27T+59T2 1 - 3.27T + 59T^{2}
61 1+13.5T+61T2 1 + 13.5T + 61T^{2}
67 1−3.52iT−67T2 1 - 3.52iT - 67T^{2}
71 1+4.54T+71T2 1 + 4.54T + 71T^{2}
73 1+6.50iT−73T2 1 + 6.50iT - 73T^{2}
79 1+7.27T+79T2 1 + 7.27T + 79T^{2}
83 1−7.40iT−83T2 1 - 7.40iT - 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

−9.384066696427408896320986229769, −8.355961166120597213748209635776, −7.65354184175091055727912419306, −7.37940209413697101786663662767, −5.97895002338454314539707554362, −5.21747600458643885160863039927, −4.12936958292740178855440717374, −2.87246584958509110039018861677, −1.57719551285500936595430011296, −0.14982028937357912319505609570, 2.36985197177523433474671115118, 3.39068386288163115165096481636, 4.40872524139374407758221726370, 4.97758471321037013244516068047, 6.33452093959358487028270113513, 7.16299970214427679415899052316, 8.113536126278013440736777859913, 8.805116315245803231078958460782, 10.05749508215314717036121780905, 10.37546714804576652590577271595

Graph of the ZZ-function along the critical line