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Properties

Label 2-85-85.84-c1-0-2
Degree 22
Conductor 8585
Sign −0.379−0.925i-0.379 - 0.925i
Analytic cond. 0.6787280.678728
Root an. cond. 0.8238490.823849
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  + 2.57i·2-s + 2.37·3-s − 4.64·4-s + (−1.95 − 1.08i)5-s + 6.12i·6-s + 1.53·7-s − 6.82i·8-s + 2.64·9-s + (2.79 − 5.04i)10-s − 3.95i·11-s − 11.0·12-s + 4.24i·13-s + 3.95i·14-s + (−4.64 − 2.57i)15-s + 8.29·16-s + (−3.21 − 2.57i)17-s + ⋯
L(s)  = 1  + 1.82i·2-s + 1.37·3-s − 2.32·4-s + (−0.874 − 0.485i)5-s + 2.50i·6-s + 0.579·7-s − 2.41i·8-s + 0.881·9-s + (0.884 − 1.59i)10-s − 1.19i·11-s − 3.18·12-s + 1.17i·13-s + 1.05i·14-s + (−1.19 − 0.665i)15-s + 2.07·16-s + (−0.780 − 0.625i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((−0.379−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((−0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: −0.379−0.925i-0.379 - 0.925i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(84,⋅)\chi_{85} (84, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), −0.379−0.925i)(2,\ 85,\ (\ :1/2),\ -0.379 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.634464+0.945503i0.634464 + 0.945503i
L(12)L(\frac12) ≈\approx 0.634464+0.945503i0.634464 + 0.945503i
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)
bad5 1+(1.95+1.08i)T 1 + (1.95 + 1.08i)T
17 1+(3.21+2.57i)T 1 + (3.21 + 2.57i)T
good2 1−2.57iT−2T2 1 - 2.57iT - 2T^{2}
3 1−2.37T+3T2 1 - 2.37T + 3T^{2}
7 1−1.53T+7T2 1 - 1.53T + 7T^{2}
11 1+3.95iT−11T2 1 + 3.95iT - 11T^{2}
13 1−4.24iT−13T2 1 - 4.24iT - 13T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+0.692T+23T2 1 + 0.692T + 23T^{2}
29 1−4.33iT−29T2 1 - 4.33iT - 29T^{2}
31 1−1.78iT−31T2 1 - 1.78iT - 31T^{2}
37 1+3.06T+37T2 1 + 3.06T + 37T^{2}
41 1+2.16iT−41T2 1 + 2.16iT - 41T^{2}
43 1+4.24iT−43T2 1 + 4.24iT - 43T^{2}
47 1−6.06iT−47T2 1 - 6.06iT - 47T^{2}
53 1+5.15iT−53T2 1 + 5.15iT - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+10.0iT−61T2 1 + 10.0iT - 61T^{2}
67 1−4.24iT−67T2 1 - 4.24iT - 67T^{2}
71 1−16.2iT−71T2 1 - 16.2iT - 71T^{2}
73 1−8.66T+73T2 1 - 8.66T + 73T^{2}
79 1+1.78iT−79T2 1 + 1.78iT - 79T^{2}
83 1+0.913iT−83T2 1 + 0.913iT - 83T^{2}
89 1−4.93T+89T2 1 - 4.93T + 89T^{2}
97 1−11.1T+97T2 1 - 11.1T + 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

−14.48162940512177847276907176916, −14.10893658439922283577666474424, −13.16754113512634457994550190849, −11.46455011086872498069795365583, −9.189790641716475196893380937277, −8.666188862532127306995089434564, −7.895799120571995791302387879684, −6.86018552501915105505399030637, −5.05152282648177571533560451126, −3.77999412062202688487984034550, 2.20995064530167775550758063233, 3.40764566604337659687836213943, 4.51755913816531217305794115046, 7.68373300389451020797009461455, 8.513038257217313726281397790273, 9.730902690072061875399874804286, 10.66881779841706634799649368121, 11.74269134574097268269674451698, 12.74068404755526734681786618546, 13.66092930278243816096946520329

Graph of the ZZ-function along the critical line