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Properties

Label 2-85-85.9-c1-0-1
Degree 22
Conductor 8585
Sign 0.0763−0.997i0.0763 - 0.997i
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  + (0.672 + 0.672i)2-s + (−0.947 + 2.28i)3-s − 1.09i·4-s + (−0.734 + 2.11i)5-s + (−2.17 + 0.901i)6-s + (0.492 − 0.204i)7-s + (2.08 − 2.08i)8-s + (−2.20 − 2.20i)9-s + (−1.91 + 0.926i)10-s + (4.31 − 1.78i)11-s + (2.50 + 1.03i)12-s − 3.92·13-s + (0.468 + 0.194i)14-s + (−4.13 − 3.68i)15-s + 0.614·16-s + (4.03 − 0.867i)17-s + ⋯
L(s)  = 1  + (0.475 + 0.475i)2-s + (−0.546 + 1.32i)3-s − 0.547i·4-s + (−0.328 + 0.944i)5-s + (−0.888 + 0.367i)6-s + (0.186 − 0.0771i)7-s + (0.736 − 0.736i)8-s + (−0.736 − 0.736i)9-s + (−0.605 + 0.293i)10-s + (1.30 − 0.539i)11-s + (0.722 + 0.299i)12-s − 1.08·13-s + (0.125 + 0.0519i)14-s + (−1.06 − 0.950i)15-s + 0.153·16-s + (0.977 − 0.210i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.0763−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.0763−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.0763−0.997i0.0763 - 0.997i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(9,⋅)\chi_{85} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.0763−0.997i)(2,\ 85,\ (\ :1/2),\ 0.0763 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.763622+0.707402i0.763622 + 0.707402i
L(12)L(\frac12) ≈\approx 0.763622+0.707402i0.763622 + 0.707402i
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+(0.734−2.11i)T 1 + (0.734 - 2.11i)T
17 1+(−4.03+0.867i)T 1 + (-4.03 + 0.867i)T
good2 1+(−0.672−0.672i)T+2iT2 1 + (-0.672 - 0.672i)T + 2iT^{2}
3 1+(0.947−2.28i)T+(−2.12−2.12i)T2 1 + (0.947 - 2.28i)T + (-2.12 - 2.12i)T^{2}
7 1+(−0.492+0.204i)T+(4.94−4.94i)T2 1 + (-0.492 + 0.204i)T + (4.94 - 4.94i)T^{2}
11 1+(−4.31+1.78i)T+(7.77−7.77i)T2 1 + (-4.31 + 1.78i)T + (7.77 - 7.77i)T^{2}
13 1+3.92T+13T2 1 + 3.92T + 13T^{2}
19 1+(−0.708+0.708i)T−19iT2 1 + (-0.708 + 0.708i)T - 19iT^{2}
23 1+(2.19+5.30i)T+(−16.2+16.2i)T2 1 + (2.19 + 5.30i)T + (-16.2 + 16.2i)T^{2}
29 1+(3.17−7.65i)T+(−20.5−20.5i)T2 1 + (3.17 - 7.65i)T + (-20.5 - 20.5i)T^{2}
31 1+(2.29+0.948i)T+(21.9+21.9i)T2 1 + (2.29 + 0.948i)T + (21.9 + 21.9i)T^{2}
37 1+(1.56−3.78i)T+(−26.1−26.1i)T2 1 + (1.56 - 3.78i)T + (-26.1 - 26.1i)T^{2}
41 1+(0.311+0.752i)T+(−28.9+28.9i)T2 1 + (0.311 + 0.752i)T + (-28.9 + 28.9i)T^{2}
43 1+(−4.55+4.55i)T−43iT2 1 + (-4.55 + 4.55i)T - 43iT^{2}
47 1−1.70T+47T2 1 - 1.70T + 47T^{2}
53 1+(−5.77−5.77i)T+53iT2 1 + (-5.77 - 5.77i)T + 53iT^{2}
59 1+(4.28+4.28i)T+59iT2 1 + (4.28 + 4.28i)T + 59iT^{2}
61 1+(0.432+1.04i)T+(−43.1+43.1i)T2 1 + (0.432 + 1.04i)T + (-43.1 + 43.1i)T^{2}
67 1+7.68iT−67T2 1 + 7.68iT - 67T^{2}
71 1+(−6.41−2.65i)T+(50.2+50.2i)T2 1 + (-6.41 - 2.65i)T + (50.2 + 50.2i)T^{2}
73 1+(11.1+4.62i)T+(51.6+51.6i)T2 1 + (11.1 + 4.62i)T + (51.6 + 51.6i)T^{2}
79 1+(1.36−0.566i)T+(55.8−55.8i)T2 1 + (1.36 - 0.566i)T + (55.8 - 55.8i)T^{2}
83 1+(−11.4−11.4i)T+83iT2 1 + (-11.4 - 11.4i)T + 83iT^{2}
89 1+12.0iT−89T2 1 + 12.0iT - 89T^{2}
97 1+(−0.210−0.0872i)T+(68.5+68.5i)T2 1 + (-0.210 - 0.0872i)T + (68.5 + 68.5i)T^{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

−14.53041613901997051727797124819, −14.18719824965621424311921444810, −12.13472388255126356960118454072, −11.02009103416678294638026362701, −10.28523257835607238839065608871, −9.334106653105715230028621366422, −7.27381461395879417982879893324, −6.08541919679992414669116616928, −4.88296498382985701553068765881, −3.70468341197974706217935604574, 1.73048144185212838484214803637, 4.05647116033177056536610726557, 5.54748654436376226200775698404, 7.25572320675905657850738942497, 7.968577534912149349699248915413, 9.518688751026352578855095712826, 11.62898185312708870127445809191, 11.97191094099371355707207114294, 12.64374735958038732871290547067, 13.54218158213958413741609741063

Graph of the ZZ-function along the critical line