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Properties

Label 2-85-85.19-c1-0-3
Degree 22
Conductor 8585
Sign 0.992−0.122i0.992 - 0.122i
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  + (0.352 − 0.352i)2-s + (0.0655 + 0.158i)3-s + 1.75i·4-s + (2.22 + 0.233i)5-s + (0.0789 + 0.0327i)6-s + (−3.57 − 1.48i)7-s + (1.32 + 1.32i)8-s + (2.10 − 2.10i)9-s + (0.867 − 0.702i)10-s + (−2.71 − 1.12i)11-s + (−0.277 + 0.114i)12-s + 0.983·13-s + (−1.78 + 0.739i)14-s + (0.108 + 0.367i)15-s − 2.56·16-s + (−3.97 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.249 − 0.249i)2-s + (0.0378 + 0.0913i)3-s + 0.875i·4-s + (0.994 + 0.104i)5-s + (0.0322 + 0.0133i)6-s + (−1.35 − 0.560i)7-s + (0.468 + 0.468i)8-s + (0.700 − 0.700i)9-s + (0.274 − 0.222i)10-s + (−0.817 − 0.338i)11-s + (−0.0799 + 0.0331i)12-s + 0.272·13-s + (−0.477 + 0.197i)14-s + (0.0281 + 0.0948i)15-s − 0.641·16-s + (−0.962 + 0.269i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.992−0.122i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.992−0.122i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.992−0.122i0.992 - 0.122i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(19,⋅)\chi_{85} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.992−0.122i)(2,\ 85,\ (\ :1/2),\ 0.992 - 0.122i)

Particular Values

L(1)L(1) ≈\approx 1.12583+0.0689475i1.12583 + 0.0689475i
L(12)L(\frac12) ≈\approx 1.12583+0.0689475i1.12583 + 0.0689475i
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+(−2.22−0.233i)T 1 + (-2.22 - 0.233i)T
17 1+(3.97−1.11i)T 1 + (3.97 - 1.11i)T
good2 1+(−0.352+0.352i)T−2iT2 1 + (-0.352 + 0.352i)T - 2iT^{2}
3 1+(−0.0655−0.158i)T+(−2.12+2.12i)T2 1 + (-0.0655 - 0.158i)T + (-2.12 + 2.12i)T^{2}
7 1+(3.57+1.48i)T+(4.94+4.94i)T2 1 + (3.57 + 1.48i)T + (4.94 + 4.94i)T^{2}
11 1+(2.71+1.12i)T+(7.77+7.77i)T2 1 + (2.71 + 1.12i)T + (7.77 + 7.77i)T^{2}
13 1−0.983T+13T2 1 - 0.983T + 13T^{2}
19 1+(1.81+1.81i)T+19iT2 1 + (1.81 + 1.81i)T + 19iT^{2}
23 1+(1.15−2.77i)T+(−16.2−16.2i)T2 1 + (1.15 - 2.77i)T + (-16.2 - 16.2i)T^{2}
29 1+(−0.210−0.507i)T+(−20.5+20.5i)T2 1 + (-0.210 - 0.507i)T + (-20.5 + 20.5i)T^{2}
31 1+(−6.98+2.89i)T+(21.9−21.9i)T2 1 + (-6.98 + 2.89i)T + (21.9 - 21.9i)T^{2}
37 1+(3.76+9.09i)T+(−26.1+26.1i)T2 1 + (3.76 + 9.09i)T + (-26.1 + 26.1i)T^{2}
41 1+(2.83−6.84i)T+(−28.9−28.9i)T2 1 + (2.83 - 6.84i)T + (-28.9 - 28.9i)T^{2}
43 1+(−5.85−5.85i)T+43iT2 1 + (-5.85 - 5.85i)T + 43iT^{2}
47 1−10.9T+47T2 1 - 10.9T + 47T^{2}
53 1+(2.53−2.53i)T−53iT2 1 + (2.53 - 2.53i)T - 53iT^{2}
59 1+(−0.216+0.216i)T−59iT2 1 + (-0.216 + 0.216i)T - 59iT^{2}
61 1+(2.60−6.28i)T+(−43.1−43.1i)T2 1 + (2.60 - 6.28i)T + (-43.1 - 43.1i)T^{2}
67 1−5.09iT−67T2 1 - 5.09iT - 67T^{2}
71 1+(3.33−1.38i)T+(50.2−50.2i)T2 1 + (3.33 - 1.38i)T + (50.2 - 50.2i)T^{2}
73 1+(−4.62+1.91i)T+(51.6−51.6i)T2 1 + (-4.62 + 1.91i)T + (51.6 - 51.6i)T^{2}
79 1+(11.4+4.76i)T+(55.8+55.8i)T2 1 + (11.4 + 4.76i)T + (55.8 + 55.8i)T^{2}
83 1+(5.74−5.74i)T−83iT2 1 + (5.74 - 5.74i)T - 83iT^{2}
89 1−13.2iT−89T2 1 - 13.2iT - 89T^{2}
97 1+(−4.18+1.73i)T+(68.5−68.5i)T2 1 + (-4.18 + 1.73i)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

−13.76370139201821994115295430954, −13.16734805802509375354958571629, −12.54889627371734169776664127169, −10.93048220805254405152833006369, −9.918971402220433003942939858254, −8.875250355760475775455998716885, −7.20526374946343897246947009936, −6.16517849060474482741770615816, −4.17277877480860633602450274244, −2.80434187622316532833073440321, 2.28553238167153591848681809117, 4.80572087540721883376827377287, 6.01793249832006547079345655237, 6.88672146216983981875664605944, 8.859276416804748016895937914233, 10.05710824637119186164645388167, 10.47096676235807926491150419323, 12.52350576580854581932063491295, 13.33072018912523887053738152390, 13.97833987045927277548935689243

Graph of the ZZ-function along the critical line