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Properties

Label 2-80-16.13-c1-0-4
Degree 22
Conductor 8080
Sign 0.753+0.657i0.753 + 0.657i
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  + (0.114 − 1.40i)2-s + (1.42 + 1.42i)3-s + (−1.97 − 0.323i)4-s + (0.707 − 0.707i)5-s + (2.16 − 1.84i)6-s − 0.690i·7-s + (−0.681 + 2.74i)8-s + 1.05i·9-s + (−0.915 − 1.07i)10-s + (−3.06 + 3.06i)11-s + (−2.34 − 3.26i)12-s + (−2.33 − 2.33i)13-s + (−0.973 − 0.0791i)14-s + 2.01·15-s + (3.79 + 1.27i)16-s − 5.28·17-s + ⋯
L(s)  = 1  + (0.0810 − 0.996i)2-s + (0.821 + 0.821i)3-s + (−0.986 − 0.161i)4-s + (0.316 − 0.316i)5-s + (0.885 − 0.752i)6-s − 0.261i·7-s + (−0.241 + 0.970i)8-s + 0.350i·9-s + (−0.289 − 0.340i)10-s + (−0.922 + 0.922i)11-s + (−0.678 − 0.943i)12-s + (−0.648 − 0.648i)13-s + (−0.260 − 0.0211i)14-s + 0.519·15-s + (0.947 + 0.318i)16-s − 1.28·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.753+0.657i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.753+0.657i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.753+0.657i0.753 + 0.657i
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), 0.753+0.657i)(2,\ 80,\ (\ :1/2),\ 0.753 + 0.657i)

Particular Values

L(1)L(1) ≈\approx 1.04236−0.390743i1.04236 - 0.390743i
L(12)L(\frac12) ≈\approx 1.04236−0.390743i1.04236 - 0.390743i
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+(−0.114+1.40i)T 1 + (-0.114 + 1.40i)T
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(−1.42−1.42i)T+3iT2 1 + (-1.42 - 1.42i)T + 3iT^{2}
7 1+0.690iT−7T2 1 + 0.690iT - 7T^{2}
11 1+(3.06−3.06i)T−11iT2 1 + (3.06 - 3.06i)T - 11iT^{2}
13 1+(2.33+2.33i)T+13iT2 1 + (2.33 + 2.33i)T + 13iT^{2}
17 1+5.28T+17T2 1 + 5.28T + 17T^{2}
19 1+(−5.38−5.38i)T+19iT2 1 + (-5.38 - 5.38i)T + 19iT^{2}
23 1+1.60iT−23T2 1 + 1.60iT - 23T^{2}
29 1+(−1.70−1.70i)T+29iT2 1 + (-1.70 - 1.70i)T + 29iT^{2}
31 1+4.69T+31T2 1 + 4.69T + 31T^{2}
37 1+(−7.89+7.89i)T−37iT2 1 + (-7.89 + 7.89i)T - 37iT^{2}
41 1+5.49iT−41T2 1 + 5.49iT - 41T^{2}
43 1+(0.256−0.256i)T−43iT2 1 + (0.256 - 0.256i)T - 43iT^{2}
47 1+4.60T+47T2 1 + 4.60T + 47T^{2}
53 1+(4.99−4.99i)T−53iT2 1 + (4.99 - 4.99i)T - 53iT^{2}
59 1+(−1.46+1.46i)T−59iT2 1 + (-1.46 + 1.46i)T - 59iT^{2}
61 1+(−9.33−9.33i)T+61iT2 1 + (-9.33 - 9.33i)T + 61iT^{2}
67 1+(1.94+1.94i)T+67iT2 1 + (1.94 + 1.94i)T + 67iT^{2}
71 1+2.32iT−71T2 1 + 2.32iT - 71T^{2}
73 1+1.29iT−73T2 1 + 1.29iT - 73T^{2}
79 1+5.01T+79T2 1 + 5.01T + 79T^{2}
83 1+(−7.30−7.30i)T+83iT2 1 + (-7.30 - 7.30i)T + 83iT^{2}
89 1−1.81iT−89T2 1 - 1.81iT - 89T^{2}
97 1−5.27T+97T2 1 - 5.27T + 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

−14.22890094039456774819182675157, −13.11467991561381190608720528795, −12.25280011233827527645767404237, −10.65405570675765892881303329849, −9.892896818146779231860846065307, −9.083974796699196057912080254561, −7.78165413865513886131663624735, −5.26688393391931109975047175565, −4.05901604608916736471959729945, −2.52433147495870099152724511179, 2.79467274517295580445009505376, 5.04124931081060413285163400043, 6.56741479125777722843350410967, 7.55123048161366281581913923317, 8.575080363596953921949685562426, 9.597245367975935464203977782933, 11.36328204350741554845725855087, 13.10816596063821793858297646821, 13.49053721839988048102278912219, 14.39272466205385451468999495655

Graph of the ZZ-function along the critical line