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Properties

Label 2-85-85.9-c1-0-4
Degree 22
Conductor 8585
Sign 0.811−0.584i0.811 - 0.584i
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  + (1.48 + 1.48i)2-s + (0.690 − 1.66i)3-s + 2.43i·4-s + (−2.22 − 0.184i)5-s + (3.51 − 1.45i)6-s + (−2.47 + 1.02i)7-s + (−0.641 + 0.641i)8-s + (−0.183 − 0.183i)9-s + (−3.04 − 3.59i)10-s + (−0.901 + 0.373i)11-s + (4.05 + 1.67i)12-s − 2.65·13-s + (−5.21 − 2.16i)14-s + (−1.84 + 3.58i)15-s + 2.95·16-s + (3.25 − 2.53i)17-s + ⋯
L(s)  = 1  + (1.05 + 1.05i)2-s + (0.398 − 0.963i)3-s + 1.21i·4-s + (−0.996 − 0.0826i)5-s + (1.43 − 0.593i)6-s + (−0.937 + 0.388i)7-s + (−0.226 + 0.226i)8-s + (−0.0613 − 0.0613i)9-s + (−0.961 − 1.13i)10-s + (−0.271 + 0.112i)11-s + (1.17 + 0.484i)12-s − 0.735·13-s + (−1.39 − 0.577i)14-s + (−0.477 + 0.926i)15-s + 0.738·16-s + (0.789 − 0.613i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.811−0.584i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.811−0.584i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.811−0.584i0.811 - 0.584i
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), 0.811−0.584i)(2,\ 85,\ (\ :1/2),\ 0.811 - 0.584i)

Particular Values

L(1)L(1) ≈\approx 1.38957+0.448544i1.38957 + 0.448544i
L(12)L(\frac12) ≈\approx 1.38957+0.448544i1.38957 + 0.448544i
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+(2.22+0.184i)T 1 + (2.22 + 0.184i)T
17 1+(−3.25+2.53i)T 1 + (-3.25 + 2.53i)T
good2 1+(−1.48−1.48i)T+2iT2 1 + (-1.48 - 1.48i)T + 2iT^{2}
3 1+(−0.690+1.66i)T+(−2.12−2.12i)T2 1 + (-0.690 + 1.66i)T + (-2.12 - 2.12i)T^{2}
7 1+(2.47−1.02i)T+(4.94−4.94i)T2 1 + (2.47 - 1.02i)T + (4.94 - 4.94i)T^{2}
11 1+(0.901−0.373i)T+(7.77−7.77i)T2 1 + (0.901 - 0.373i)T + (7.77 - 7.77i)T^{2}
13 1+2.65T+13T2 1 + 2.65T + 13T^{2}
19 1+(0.478−0.478i)T−19iT2 1 + (0.478 - 0.478i)T - 19iT^{2}
23 1+(−1.54−3.72i)T+(−16.2+16.2i)T2 1 + (-1.54 - 3.72i)T + (-16.2 + 16.2i)T^{2}
29 1+(−3.83+9.26i)T+(−20.5−20.5i)T2 1 + (-3.83 + 9.26i)T + (-20.5 - 20.5i)T^{2}
31 1+(9.10+3.77i)T+(21.9+21.9i)T2 1 + (9.10 + 3.77i)T + (21.9 + 21.9i)T^{2}
37 1+(2.61−6.31i)T+(−26.1−26.1i)T2 1 + (2.61 - 6.31i)T + (-26.1 - 26.1i)T^{2}
41 1+(−2.80−6.76i)T+(−28.9+28.9i)T2 1 + (-2.80 - 6.76i)T + (-28.9 + 28.9i)T^{2}
43 1+(6.29−6.29i)T−43iT2 1 + (6.29 - 6.29i)T - 43iT^{2}
47 1−0.874T+47T2 1 - 0.874T + 47T^{2}
53 1+(2.00+2.00i)T+53iT2 1 + (2.00 + 2.00i)T + 53iT^{2}
59 1+(7.59+7.59i)T+59iT2 1 + (7.59 + 7.59i)T + 59iT^{2}
61 1+(−0.793−1.91i)T+(−43.1+43.1i)T2 1 + (-0.793 - 1.91i)T + (-43.1 + 43.1i)T^{2}
67 1+5.73iT−67T2 1 + 5.73iT - 67T^{2}
71 1+(−1.03−0.430i)T+(50.2+50.2i)T2 1 + (-1.03 - 0.430i)T + (50.2 + 50.2i)T^{2}
73 1+(−2.77−1.14i)T+(51.6+51.6i)T2 1 + (-2.77 - 1.14i)T + (51.6 + 51.6i)T^{2}
79 1+(0.377−0.156i)T+(55.8−55.8i)T2 1 + (0.377 - 0.156i)T + (55.8 - 55.8i)T^{2}
83 1+(4.64+4.64i)T+83iT2 1 + (4.64 + 4.64i)T + 83iT^{2}
89 1+5.62iT−89T2 1 + 5.62iT - 89T^{2}
97 1+(−14.2−5.89i)T+(68.5+68.5i)T2 1 + (-14.2 - 5.89i)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

−14.37876706353740254326724071275, −13.26333717643990101547181542283, −12.69082900241611087180198253758, −11.78181956390120552672479861346, −9.762949362875318531403801530552, −7.972894800474475780992747159097, −7.39930671162935903341873622925, −6.32171534576200659922573144782, −4.82063475491863613291608046825, −3.18027730371528488875196874763, 3.17106105126081210182067186649, 3.85651403176183235577055423608, 5.07691285919994236489658217944, 7.15904138006511970778056030842, 8.864911233555864448888622015249, 10.30321366215386142747813516720, 10.75770521565847556304430441887, 12.35324468261987737993712564810, 12.67405243118175167539523036655, 14.24757545612069675488364941527

Graph of the ZZ-function along the critical line