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Properties

Label 2-85-17.9-c1-0-1
Degree 22
Conductor 8585
Sign 0.925−0.377i0.925 - 0.377i
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  + (0.213 − 0.213i)2-s + (0.980 + 0.406i)3-s + 1.90i·4-s + (−0.382 + 0.923i)5-s + (0.295 − 0.122i)6-s + (−0.960 − 2.31i)7-s + (0.833 + 0.833i)8-s + (−1.32 − 1.32i)9-s + (0.115 + 0.278i)10-s + (2.25 − 0.935i)11-s + (−0.775 + 1.87i)12-s − 5.61i·13-s + (−0.699 − 0.289i)14-s + (−0.750 + 0.750i)15-s − 3.46·16-s + (2.76 + 3.06i)17-s + ⋯
L(s)  = 1  + (0.150 − 0.150i)2-s + (0.565 + 0.234i)3-s + 0.954i·4-s + (−0.171 + 0.413i)5-s + (0.120 − 0.0500i)6-s + (−0.363 − 0.876i)7-s + (0.294 + 0.294i)8-s + (−0.441 − 0.441i)9-s + (0.0365 + 0.0881i)10-s + (0.681 − 0.282i)11-s + (−0.223 + 0.540i)12-s − 1.55i·13-s + (−0.186 − 0.0774i)14-s + (−0.193 + 0.193i)15-s − 0.865·16-s + (0.669 + 0.742i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.925−0.377i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.925−0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.925−0.377i0.925 - 0.377i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(26,⋅)\chi_{85} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.925−0.377i)(2,\ 85,\ (\ :1/2),\ 0.925 - 0.377i)

Particular Values

L(1)L(1) ≈\approx 1.11238+0.218319i1.11238 + 0.218319i
L(12)L(\frac12) ≈\approx 1.11238+0.218319i1.11238 + 0.218319i
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+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
17 1+(−2.76−3.06i)T 1 + (-2.76 - 3.06i)T
good2 1+(−0.213+0.213i)T−2iT2 1 + (-0.213 + 0.213i)T - 2iT^{2}
3 1+(−0.980−0.406i)T+(2.12+2.12i)T2 1 + (-0.980 - 0.406i)T + (2.12 + 2.12i)T^{2}
7 1+(0.960+2.31i)T+(−4.94+4.94i)T2 1 + (0.960 + 2.31i)T + (-4.94 + 4.94i)T^{2}
11 1+(−2.25+0.935i)T+(7.77−7.77i)T2 1 + (-2.25 + 0.935i)T + (7.77 - 7.77i)T^{2}
13 1+5.61iT−13T2 1 + 5.61iT - 13T^{2}
19 1+(5.04−5.04i)T−19iT2 1 + (5.04 - 5.04i)T - 19iT^{2}
23 1+(−0.795+0.329i)T+(16.2−16.2i)T2 1 + (-0.795 + 0.329i)T + (16.2 - 16.2i)T^{2}
29 1+(1.43−3.46i)T+(−20.5−20.5i)T2 1 + (1.43 - 3.46i)T + (-20.5 - 20.5i)T^{2}
31 1+(2.07+0.860i)T+(21.9+21.9i)T2 1 + (2.07 + 0.860i)T + (21.9 + 21.9i)T^{2}
37 1+(−4.71−1.95i)T+(26.1+26.1i)T2 1 + (-4.71 - 1.95i)T + (26.1 + 26.1i)T^{2}
41 1+(−4.72−11.4i)T+(−28.9+28.9i)T2 1 + (-4.72 - 11.4i)T + (-28.9 + 28.9i)T^{2}
43 1+(1.85+1.85i)T+43iT2 1 + (1.85 + 1.85i)T + 43iT^{2}
47 1+2.30iT−47T2 1 + 2.30iT - 47T^{2}
53 1+(−1.96+1.96i)T−53iT2 1 + (-1.96 + 1.96i)T - 53iT^{2}
59 1+(5.26+5.26i)T+59iT2 1 + (5.26 + 5.26i)T + 59iT^{2}
61 1+(0.346+0.837i)T+(−43.1+43.1i)T2 1 + (0.346 + 0.837i)T + (-43.1 + 43.1i)T^{2}
67 1+6.69T+67T2 1 + 6.69T + 67T^{2}
71 1+(−0.222−0.0921i)T+(50.2+50.2i)T2 1 + (-0.222 - 0.0921i)T + (50.2 + 50.2i)T^{2}
73 1+(2.47−5.96i)T+(−51.6−51.6i)T2 1 + (2.47 - 5.96i)T + (-51.6 - 51.6i)T^{2}
79 1+(−13.5+5.62i)T+(55.8−55.8i)T2 1 + (-13.5 + 5.62i)T + (55.8 - 55.8i)T^{2}
83 1+(9.82−9.82i)T−83iT2 1 + (9.82 - 9.82i)T - 83iT^{2}
89 1−0.395iT−89T2 1 - 0.395iT - 89T^{2}
97 1+(−3.42+8.27i)T+(−68.5−68.5i)T2 1 + (-3.42 + 8.27i)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

−14.39141701119390157032858222471, −13.16919810305998613209454708086, −12.36674382202784426789171796136, −11.05814860759571933107578131959, −9.999887615695134279543608455228, −8.500528347406063545162550398691, −7.71825536198955041337226755347, −6.22325757238837303957906756774, −3.91157390401259137900895050355, −3.18196813149302897793044728956, 2.20005399702633186438631002513, 4.52284321921065619763693254788, 5.90954251001310487501644889958, 7.16731934356936293449752271708, 8.965573362980959817892945584535, 9.311363724320658618405204762104, 11.04176385105724015416144635446, 12.06357102849648184375988677131, 13.38729402654983503156907229571, 14.24802959558016495347739827000

Graph of the ZZ-function along the critical line