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Properties

Label 2-85-5.4-c1-0-1
Degree 22
Conductor 8585
Sign 0.997−0.0654i0.997 - 0.0654i
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  − 1.57i·2-s + 2.87i·3-s − 0.494·4-s + (0.146 + 2.23i)5-s + 4.53·6-s − 1.42i·7-s − 2.37i·8-s − 5.24·9-s + (3.52 − 0.231i)10-s + 0.0740·11-s − 1.42i·12-s − 5.70i·13-s − 2.24·14-s + (−6.40 + 0.420i)15-s − 4.74·16-s + i·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + 1.65i·3-s − 0.247·4-s + (0.0654 + 0.997i)5-s + 1.85·6-s − 0.536i·7-s − 0.840i·8-s − 1.74·9-s + (1.11 − 0.0731i)10-s + 0.0223·11-s − 0.410i·12-s − 1.58i·13-s − 0.599·14-s + (−1.65 + 0.108i)15-s − 1.18·16-s + 0.242i·17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.997−0.0654i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.997−0.0654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.997−0.0654i0.997 - 0.0654i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(69,⋅)\chi_{85} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.997−0.0654i)(2,\ 85,\ (\ :1/2),\ 0.997 - 0.0654i)

Particular Values

L(1)L(1) ≈\approx 1.01553+0.0332860i1.01553 + 0.0332860i
L(12)L(\frac12) ≈\approx 1.01553+0.0332860i1.01553 + 0.0332860i
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+(−0.146−2.23i)T 1 + (-0.146 - 2.23i)T
17 1−iT 1 - iT
good2 1+1.57iT−2T2 1 + 1.57iT - 2T^{2}
3 1−2.87iT−3T2 1 - 2.87iT - 3T^{2}
7 1+1.42iT−7T2 1 + 1.42iT - 7T^{2}
11 1−0.0740T+11T2 1 - 0.0740T + 11T^{2}
13 1+5.70iT−13T2 1 + 5.70iT - 13T^{2}
19 1−4.90T+19T2 1 - 4.90T + 19T^{2}
23 1−3.88iT−23T2 1 - 3.88iT - 23T^{2}
29 1+5.91T+29T2 1 + 5.91T + 29T^{2}
31 1−0.388T+31T2 1 - 0.388T + 31T^{2}
37 1+9.91iT−37T2 1 + 9.91iT - 37T^{2}
41 1+6.61T+41T2 1 + 6.61T + 41T^{2}
43 1−6.94iT−43T2 1 - 6.94iT - 43T^{2}
47 1−5.70iT−47T2 1 - 5.70iT - 47T^{2}
53 1+0.0216iT−53T2 1 + 0.0216iT - 53T^{2}
59 1−2T+59T2 1 - 2T + 59T^{2}
61 1+3.47T+61T2 1 + 3.47T + 61T^{2}
67 1−6.71iT−67T2 1 - 6.71iT - 67T^{2}
71 1−3.84T+71T2 1 - 3.84T + 71T^{2}
73 1−13.5iT−73T2 1 - 13.5iT - 73T^{2}
79 1−1.05T+79T2 1 - 1.05T + 79T^{2}
83 1+11.8iT−83T2 1 + 11.8iT - 83T^{2}
89 1−1.99T+89T2 1 - 1.99T + 89T^{2}
97 1+5.34iT−97T2 1 + 5.34iT - 97T^{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

−14.39290144291424038432032453265, −13.10973978871794805504059537895, −11.55171301052548810113526768587, −10.80529398385985662947645173805, −10.19609308384168762839618956373, −9.451939587445893698388504674552, −7.49911545730733127152346504893, −5.60250454769964844469220937588, −3.83429780120465818778217711126, −3.05165080139244125986926659785, 1.93659467977269682117792101212, 5.15909887959126639872194083157, 6.32858783174428522241835341773, 7.21972602277824192909256680897, 8.281939640133313509184731407968, 9.135930178200450976256179791144, 11.67804568240630967046975780479, 12.09264939637504968630227480861, 13.45918334354064281070858125900, 14.01984473898385951141564077387

Graph of the ZZ-function along the critical line