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Properties

Label 2-80-5.4-c1-0-1
Degree 22
Conductor 8080
Sign 0.447+0.894i0.447 + 0.894i
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  − 2i·3-s + (−1 − 2i)5-s + 2i·7-s − 9-s + 4·11-s + 4i·13-s + (−4 + 2i)15-s − 4·19-s + 4·21-s + 2i·23-s + (−3 + 4i)25-s − 4i·27-s − 2·29-s − 8i·33-s + (4 − 2i)35-s + ⋯
L(s)  = 1  − 1.15i·3-s + (−0.447 − 0.894i)5-s + 0.755i·7-s − 0.333·9-s + 1.20·11-s + 1.10i·13-s + (−1.03 + 0.516i)15-s − 0.917·19-s + 0.872·21-s + 0.417i·23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s − 0.371·29-s − 1.39i·33-s + (0.676 − 0.338i)35-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(49,⋅)\chi_{80} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.447+0.894i)(2,\ 80,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.789428−0.487893i0.789428 - 0.487893i
L(12)L(\frac12) ≈\approx 0.789428−0.487893i0.789428 - 0.487893i
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+2i)T 1 + (1 + 2i)T
good3 1+2iT−3T2 1 + 2iT - 3T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+4iT−37T2 1 + 4iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1+6iT−47T2 1 + 6iT - 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1−14iT−67T2 1 - 14iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−8iT−73T2 1 - 8iT - 73T^{2}
79 1−16T+79T2 1 - 16T + 79T^{2}
83 1+2iT−83T2 1 + 2iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+16iT−97T2 1 + 16iT - 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

−14.03027895351106023041294264280, −12.91997052424831310620284565891, −12.15060959609603369715962295845, −11.46522605854082072550973007185, −9.352390215483697261062375577735, −8.531576274232393632586704936129, −7.21583158999792306410434978903, −6.06383267017174124142765982170, −4.26729765342000317017177688222, −1.71634013614640719267961657709, 3.43992358269873926659198241096, 4.44387010686589718708577246840, 6.36127294369327897367988140029, 7.67898330238913181870458570066, 9.210189714894171573445856454892, 10.44721562698896132870423004877, 10.86241726330800794944868649596, 12.27661252309399260434141940896, 13.80403053219716984980398795584, 14.87516829368151963944878567818

Graph of the ZZ-function along the critical line