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Properties

Label 2-80-16.13-c1-0-1
Degree 22
Conductor 8080
Sign −0.0988−0.995i-0.0988 - 0.995i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
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  + (−0.376 + 1.36i)2-s + (1.82 + 1.82i)3-s + (−1.71 − 1.02i)4-s + (−0.707 + 0.707i)5-s + (−3.18 + 1.80i)6-s − 4.50i·7-s + (2.04 − 1.95i)8-s + 3.68i·9-s + (−0.697 − 1.23i)10-s + (−1.64 + 1.64i)11-s + (−1.25 − 5.01i)12-s + (1.51 + 1.51i)13-s + (6.14 + 1.69i)14-s − 2.58·15-s + (1.88 + 3.52i)16-s + 1.45·17-s + ⋯
L(s)  = 1  + (−0.266 + 0.963i)2-s + (1.05 + 1.05i)3-s + (−0.857 − 0.513i)4-s + (−0.316 + 0.316i)5-s + (−1.29 + 0.735i)6-s − 1.70i·7-s + (0.723 − 0.689i)8-s + 1.22i·9-s + (−0.220 − 0.389i)10-s + (−0.494 + 0.494i)11-s + (−0.363 − 1.44i)12-s + (0.421 + 0.421i)13-s + (1.64 + 0.454i)14-s − 0.667·15-s + (0.472 + 0.881i)16-s + 0.353·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((−0.0988−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0988 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((−0.0988−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0988 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −0.0988−0.995i-0.0988 - 0.995i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(61,⋅)\chi_{80} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), −0.0988−0.995i)(2,\ 80,\ (\ :1/2),\ -0.0988 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.666421+0.735884i0.666421 + 0.735884i
L(12)L(\frac12) ≈\approx 0.666421+0.735884i0.666421 + 0.735884i
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+(0.376−1.36i)T 1 + (0.376 - 1.36i)T
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(−1.82−1.82i)T+3iT2 1 + (-1.82 - 1.82i)T + 3iT^{2}
7 1+4.50iT−7T2 1 + 4.50iT - 7T^{2}
11 1+(1.64−1.64i)T−11iT2 1 + (1.64 - 1.64i)T - 11iT^{2}
13 1+(−1.51−1.51i)T+13iT2 1 + (-1.51 - 1.51i)T + 13iT^{2}
17 1−1.45T+17T2 1 - 1.45T + 17T^{2}
19 1+(2.67+2.67i)T+19iT2 1 + (2.67 + 2.67i)T + 19iT^{2}
23 1+2.37iT−23T2 1 + 2.37iT - 23T^{2}
29 1+(−0.924−0.924i)T+29iT2 1 + (-0.924 - 0.924i)T + 29iT^{2}
31 1+7.20T+31T2 1 + 7.20T + 31T^{2}
37 1+(5.21−5.21i)T−37iT2 1 + (5.21 - 5.21i)T - 37iT^{2}
41 1−6.41iT−41T2 1 - 6.41iT - 41T^{2}
43 1+(−7.65+7.65i)T−43iT2 1 + (-7.65 + 7.65i)T - 43iT^{2}
47 1+2.51T+47T2 1 + 2.51T + 47T^{2}
53 1+(−1.50+1.50i)T−53iT2 1 + (-1.50 + 1.50i)T - 53iT^{2}
59 1+(5.31−5.31i)T−59iT2 1 + (5.31 - 5.31i)T - 59iT^{2}
61 1+(1.02+1.02i)T+61iT2 1 + (1.02 + 1.02i)T + 61iT^{2}
67 1+(−5.22−5.22i)T+67iT2 1 + (-5.22 - 5.22i)T + 67iT^{2}
71 1−1.92iT−71T2 1 - 1.92iT - 71T^{2}
73 1−1.39iT−73T2 1 - 1.39iT - 73T^{2}
79 1−5.06T+79T2 1 - 5.06T + 79T^{2}
83 1+(2.44+2.44i)T+83iT2 1 + (2.44 + 2.44i)T + 83iT^{2}
89 1+9.36iT−89T2 1 + 9.36iT - 89T^{2}
97 1−18.6T+97T2 1 - 18.6T + 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

−14.71796274625662716417148016335, −14.06951415423938629829684853257, −13.14877535830764258062259869839, −10.72763045386641419368718933036, −10.13652678341726017087153635563, −8.970342272924689566178977452773, −7.84126496234031060278032406163, −6.86794620573944544649623891067, −4.65335156379218792221861664415, −3.73755730382643648849698937949, 2.03022076746033015488101457574, 3.29575440362236288092380560256, 5.60535764418082492606475597258, 7.78351584323572552141405667020, 8.527250540411935600621316309651, 9.269412083593274534852182304164, 11.01396473911854365108775664753, 12.37571028843833352734430446061, 12.66316755462734607074089909619, 13.81233730874694649503616323634

Graph of the ZZ-function along the critical line