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Properties

Label 2-80-80.29-c1-0-9
Degree 22
Conductor 8080
Sign −0.168+0.985i-0.168 + 0.985i
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  + (−0.161 − 1.40i)2-s + (0.734 − 0.734i)3-s + (−1.94 + 0.452i)4-s + (−1.17 − 1.90i)5-s + (−1.14 − 0.913i)6-s + 1.71·7-s + (0.949 + 2.66i)8-s + 1.92i·9-s + (−2.48 + 1.95i)10-s + (2.82 − 2.82i)11-s + (−1.09 + 1.76i)12-s + (−2.59 + 2.59i)13-s + (−0.276 − 2.40i)14-s + (−2.25 − 0.537i)15-s + (3.59 − 1.76i)16-s + 1.89i·17-s + ⋯
L(s)  = 1  + (−0.113 − 0.993i)2-s + (0.423 − 0.423i)3-s + (−0.974 + 0.226i)4-s + (−0.524 − 0.851i)5-s + (−0.469 − 0.372i)6-s + 0.648·7-s + (0.335 + 0.941i)8-s + 0.640i·9-s + (−0.786 + 0.617i)10-s + (0.852 − 0.852i)11-s + (−0.317 + 0.508i)12-s + (−0.719 + 0.719i)13-s + (−0.0738 − 0.643i)14-s + (−0.583 − 0.138i)15-s + (0.897 − 0.440i)16-s + 0.460i·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((−0.168+0.985i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((−0.168+0.985i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −0.168+0.985i-0.168 + 0.985i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(29,⋅)\chi_{80} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), −0.168+0.985i)(2,\ 80,\ (\ :1/2),\ -0.168 + 0.985i)

Particular Values

L(1)L(1) ≈\approx 0.593411−0.703776i0.593411 - 0.703776i
L(12)L(\frac12) ≈\approx 0.593411−0.703776i0.593411 - 0.703776i
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+(0.161+1.40i)T 1 + (0.161 + 1.40i)T
5 1+(1.17+1.90i)T 1 + (1.17 + 1.90i)T
good3 1+(−0.734+0.734i)T−3iT2 1 + (-0.734 + 0.734i)T - 3iT^{2}
7 1−1.71T+7T2 1 - 1.71T + 7T^{2}
11 1+(−2.82+2.82i)T−11iT2 1 + (-2.82 + 2.82i)T - 11iT^{2}
13 1+(2.59−2.59i)T−13iT2 1 + (2.59 - 2.59i)T - 13iT^{2}
17 1−1.89iT−17T2 1 - 1.89iT - 17T^{2}
19 1+(−2.89−2.89i)T+19iT2 1 + (-2.89 - 2.89i)T + 19iT^{2}
23 1−2.00T+23T2 1 - 2.00T + 23T^{2}
29 1+(6.72+6.72i)T+29iT2 1 + (6.72 + 6.72i)T + 29iT^{2}
31 1+7.11T+31T2 1 + 7.11T + 31T^{2}
37 1+(−2.25−2.25i)T+37iT2 1 + (-2.25 - 2.25i)T + 37iT^{2}
41 1−1.59iT−41T2 1 - 1.59iT - 41T^{2}
43 1+(8.06+8.06i)T+43iT2 1 + (8.06 + 8.06i)T + 43iT^{2}
47 1−4.43iT−47T2 1 - 4.43iT - 47T^{2}
53 1+(−0.481−0.481i)T+53iT2 1 + (-0.481 - 0.481i)T + 53iT^{2}
59 1+(−3.08+3.08i)T−59iT2 1 + (-3.08 + 3.08i)T - 59iT^{2}
61 1+(−3.46−3.46i)T+61iT2 1 + (-3.46 - 3.46i)T + 61iT^{2}
67 1+(−1.80+1.80i)T−67iT2 1 + (-1.80 + 1.80i)T - 67iT^{2}
71 1−0.379iT−71T2 1 - 0.379iT - 71T^{2}
73 1−8.37T+73T2 1 - 8.37T + 73T^{2}
79 1−11.2T+79T2 1 - 11.2T + 79T^{2}
83 1+(8.24−8.24i)T−83iT2 1 + (8.24 - 8.24i)T - 83iT^{2}
89 1+11.9iT−89T2 1 + 11.9iT - 89T^{2}
97 1−6.50iT−97T2 1 - 6.50iT - 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

−13.85683644678238828575415444092, −12.93114899151371344449372519681, −11.82956232471594561059082557174, −11.15559107902226723162603906496, −9.513756907126071962480608850244, −8.514573610283005270587140680518, −7.66126779756824784354080325840, −5.20701664578464931718198818531, −3.84069826645771333305387753811, −1.71650780813667433124270790332, 3.57954630959318027825433039589, 5.01099989333800356264361893145, 6.82842522520423858023852797214, 7.60831575936211312757661852472, 9.035842943502200819094537211908, 9.914829144796099010591664576025, 11.36547128909570204310854211682, 12.69351780976896352691880962471, 14.31051691936246715757610062454, 14.81509773716136642927017166041

Graph of the ZZ-function along the critical line