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Properties

Label 2-85-17.16-c1-0-0
Degree 22
Conductor 8585
Sign 0.615−0.787i0.615 - 0.787i
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  − 2.17·2-s + 0.539i·3-s + 2.70·4-s − i·5-s − 1.17i·6-s + 4.87i·7-s − 1.53·8-s + 2.70·9-s + 2.17i·10-s + 3.17i·11-s + 1.46i·12-s + 2.63·13-s − 10.5i·14-s + 0.539·15-s − 2.07·16-s + (−3.24 − 2.53i)17-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.311i·3-s + 1.35·4-s − 0.447i·5-s − 0.477i·6-s + 1.84i·7-s − 0.544·8-s + 0.903·9-s + 0.686i·10-s + 0.955i·11-s + 0.421i·12-s + 0.729·13-s − 2.82i·14-s + 0.139·15-s − 0.519·16-s + (−0.787 − 0.615i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.615−0.787i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.615−0.787i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.615−0.787i0.615 - 0.787i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(16,⋅)\chi_{85} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.615−0.787i)(2,\ 85,\ (\ :1/2),\ 0.615 - 0.787i)

Particular Values

L(1)L(1) ≈\approx 0.455653+0.222171i0.455653 + 0.222171i
L(12)L(\frac12) ≈\approx 0.455653+0.222171i0.455653 + 0.222171i
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+iT 1 + iT
17 1+(3.24+2.53i)T 1 + (3.24 + 2.53i)T
good2 1+2.17T+2T2 1 + 2.17T + 2T^{2}
3 1−0.539iT−3T2 1 - 0.539iT - 3T^{2}
7 1−4.87iT−7T2 1 - 4.87iT - 7T^{2}
11 1−3.17iT−11T2 1 - 3.17iT - 11T^{2}
13 1−2.63T+13T2 1 - 2.63T + 13T^{2}
19 1−1.07T+19T2 1 - 1.07T + 19T^{2}
23 1+5.21iT−23T2 1 + 5.21iT - 23T^{2}
29 1−2.92iT−29T2 1 - 2.92iT - 29T^{2}
31 1+4.09iT−31T2 1 + 4.09iT - 31T^{2}
37 1+5.26iT−37T2 1 + 5.26iT - 37T^{2}
41 1−5.60iT−41T2 1 - 5.60iT - 41T^{2}
43 1−3.36T+43T2 1 - 3.36T + 43T^{2}
47 1+6.78T+47T2 1 + 6.78T + 47T^{2}
53 1−3.75T+53T2 1 - 3.75T + 53T^{2}
59 1+2.34T+59T2 1 + 2.34T + 59T^{2}
61 1+12.2iT−61T2 1 + 12.2iT - 61T^{2}
67 1−10.2T+67T2 1 - 10.2T + 67T^{2}
71 1+4.06iT−71T2 1 + 4.06iT - 71T^{2}
73 1+11.0iT−73T2 1 + 11.0iT - 73T^{2}
79 1+6.92iT−79T2 1 + 6.92iT - 79T^{2}
83 1−8.23T+83T2 1 - 8.23T + 83T^{2}
89 1−7.15T+89T2 1 - 7.15T + 89T^{2}
97 1−8.18iT−97T2 1 - 8.18iT - 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

−14.95296491144703727800965362269, −13.02572931974601969823710537263, −12.04275257309838249969100381306, −10.87138738782782313888309989145, −9.544123839418434955514328880299, −9.103096593238461609267116044419, −8.017770023677950638430227825144, −6.56523488289068496718447896444, −4.81491545421468211833334731675, −2.07961361918891517013062975765, 1.21563611534517583450234751434, 3.91769165052456558310298023946, 6.60113214829784771655336847090, 7.38525042743959833248473495426, 8.373305896050367527806367713962, 9.824598648838181641358882583407, 10.62113558302734283458921638363, 11.29900725424366086247172431906, 13.29713450455463627154021523586, 13.83206094318183029450694449026

Graph of the ZZ-function along the critical line