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Properties

Label 2-85-17.8-c1-0-1
Degree 22
Conductor 8585
Sign 0.711−0.702i0.711 - 0.702i
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  + (0.680 − 0.680i)2-s + (−1.01 + 2.44i)3-s + 1.07i·4-s + (−0.923 − 0.382i)5-s + (0.976 + 2.35i)6-s + (2.85 − 1.18i)7-s + (2.09 + 2.09i)8-s + (−2.84 − 2.84i)9-s + (−0.889 + 0.368i)10-s + (−2.34 − 5.66i)11-s + (−2.62 − 1.08i)12-s + 1.16i·13-s + (1.14 − 2.75i)14-s + (1.87 − 1.87i)15-s + 0.703·16-s + (1.25 + 3.92i)17-s + ⋯
L(s)  = 1  + (0.481 − 0.481i)2-s + (−0.585 + 1.41i)3-s + 0.536i·4-s + (−0.413 − 0.171i)5-s + (0.398 + 0.962i)6-s + (1.08 − 0.447i)7-s + (0.739 + 0.739i)8-s + (−0.946 − 0.946i)9-s + (−0.281 + 0.116i)10-s + (−0.707 − 1.70i)11-s + (−0.757 − 0.313i)12-s + 0.321i·13-s + (0.304 − 0.735i)14-s + (0.483 − 0.483i)15-s + 0.175·16-s + (0.305 + 0.952i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.711−0.702i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.711−0.702i0.711 - 0.702i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(76,⋅)\chi_{85} (76, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.711−0.702i)(2,\ 85,\ (\ :1/2),\ 0.711 - 0.702i)

Particular Values

L(1)L(1) ≈\approx 0.967845+0.397114i0.967845 + 0.397114i
L(12)L(\frac12) ≈\approx 0.967845+0.397114i0.967845 + 0.397114i
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+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
17 1+(−1.25−3.92i)T 1 + (-1.25 - 3.92i)T
good2 1+(−0.680+0.680i)T−2iT2 1 + (-0.680 + 0.680i)T - 2iT^{2}
3 1+(1.01−2.44i)T+(−2.12−2.12i)T2 1 + (1.01 - 2.44i)T + (-2.12 - 2.12i)T^{2}
7 1+(−2.85+1.18i)T+(4.94−4.94i)T2 1 + (-2.85 + 1.18i)T + (4.94 - 4.94i)T^{2}
11 1+(2.34+5.66i)T+(−7.77+7.77i)T2 1 + (2.34 + 5.66i)T + (-7.77 + 7.77i)T^{2}
13 1−1.16iT−13T2 1 - 1.16iT - 13T^{2}
19 1+(−3.83+3.83i)T−19iT2 1 + (-3.83 + 3.83i)T - 19iT^{2}
23 1+(1.19+2.88i)T+(−16.2+16.2i)T2 1 + (1.19 + 2.88i)T + (-16.2 + 16.2i)T^{2}
29 1+(4.61+1.91i)T+(20.5+20.5i)T2 1 + (4.61 + 1.91i)T + (20.5 + 20.5i)T^{2}
31 1+(1.42−3.44i)T+(−21.9−21.9i)T2 1 + (1.42 - 3.44i)T + (-21.9 - 21.9i)T^{2}
37 1+(0.151−0.366i)T+(−26.1−26.1i)T2 1 + (0.151 - 0.366i)T + (-26.1 - 26.1i)T^{2}
41 1+(−1.57+0.651i)T+(28.9−28.9i)T2 1 + (-1.57 + 0.651i)T + (28.9 - 28.9i)T^{2}
43 1+(−0.0189−0.0189i)T+43iT2 1 + (-0.0189 - 0.0189i)T + 43iT^{2}
47 1+5.43iT−47T2 1 + 5.43iT - 47T^{2}
53 1+(0.244−0.244i)T−53iT2 1 + (0.244 - 0.244i)T - 53iT^{2}
59 1+(−2.87−2.87i)T+59iT2 1 + (-2.87 - 2.87i)T + 59iT^{2}
61 1+(11.4−4.76i)T+(43.1−43.1i)T2 1 + (11.4 - 4.76i)T + (43.1 - 43.1i)T^{2}
67 1+5.62T+67T2 1 + 5.62T + 67T^{2}
71 1+(−4.12+9.95i)T+(−50.2−50.2i)T2 1 + (-4.12 + 9.95i)T + (-50.2 - 50.2i)T^{2}
73 1+(1.52+0.633i)T+(51.6+51.6i)T2 1 + (1.52 + 0.633i)T + (51.6 + 51.6i)T^{2}
79 1+(−2.01−4.87i)T+(−55.8+55.8i)T2 1 + (-2.01 - 4.87i)T + (-55.8 + 55.8i)T^{2}
83 1+(8.78−8.78i)T−83iT2 1 + (8.78 - 8.78i)T - 83iT^{2}
89 1−3.22iT−89T2 1 - 3.22iT - 89T^{2}
97 1+(−11.9−4.94i)T+(68.5+68.5i)T2 1 + (-11.9 - 4.94i)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

−14.31975840917048058449574331719, −13.34387287413421838893300974090, −11.85632119275467760944128780948, −11.08583476200567583409284123998, −10.61312727147919754163920145927, −8.811215207407080942777835111779, −7.81049110404248174468690449369, −5.48127203637532832781796205064, −4.48111923211612207638936495720, −3.42465488065181914391997182381, 1.75996424768553766619335775501, 4.88691117773629465018555276169, 5.79730450156628542030604841184, 7.38813230527674191610430998057, 7.61134984261568230546918830825, 9.822738169103701591085694778603, 11.24435158719900455481647427100, 12.14577537482450351078195706841, 13.02288874751909755927698994809, 14.17121348883406447198866836789

Graph of the ZZ-function along the critical line