Properties

Label 2-85-85.9-c1-0-2
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/2ΓC(s)L(s)=((0.992+0.122i)Λ(2s)\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/2ΓC(s+1/2)L(s)=((0.992+0.122i)Λ(1s)\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    =    5175 \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(9,)\chi_{85} (9, \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 1.125830.0689475i1.12583 - 0.0689475i
L(12)L(\frac12) \approx 1.125830.0689475i1.12583 - 0.0689475i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+(0.3520.352i)T+2iT2 1 + (-0.352 - 0.352i)T + 2iT^{2}
3 1+(0.0655+0.158i)T+(2.122.12i)T2 1 + (-0.0655 + 0.158i)T + (-2.12 - 2.12i)T^{2}
7 1+(3.571.48i)T+(4.944.94i)T2 1 + (3.57 - 1.48i)T + (4.94 - 4.94i)T^{2}
11 1+(2.711.12i)T+(7.777.77i)T2 1 + (2.71 - 1.12i)T + (7.77 - 7.77i)T^{2}
13 10.983T+13T2 1 - 0.983T + 13T^{2}
19 1+(1.811.81i)T19iT2 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+(20.520.5i)T2 1 + (-0.210 + 0.507i)T + (-20.5 - 20.5i)T^{2}
31 1+(6.982.89i)T+(21.9+21.9i)T2 1 + (-6.98 - 2.89i)T + (21.9 + 21.9i)T^{2}
37 1+(3.769.09i)T+(26.126.1i)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)T43iT2 1 + (-5.85 + 5.85i)T - 43iT^{2}
47 110.9T+47T2 1 - 10.9T + 47T^{2}
53 1+(2.53+2.53i)T+53iT2 1 + (2.53 + 2.53i)T + 53iT^{2}
59 1+(0.2160.216i)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+5.09iT67T2 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+(4.621.91i)T+(51.6+51.6i)T2 1 + (-4.62 - 1.91i)T + (51.6 + 51.6i)T^{2}
79 1+(11.44.76i)T+(55.855.8i)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+13.2iT89T2 1 + 13.2iT - 89T^{2}
97 1+(4.181.73i)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(1αj,pps)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

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

Graph of the ZZ-function along the critical line