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Properties

Label 2-850-85.12-c1-0-3
Degree 22
Conductor 850850
Sign −0.918+0.396i-0.918 + 0.396i
Analytic cond. 6.787286.78728
Root an. cond. 2.605242.60524
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  + (−0.382 + 0.923i)2-s + (−0.372 + 1.87i)3-s + (−0.707 − 0.707i)4-s + (−1.58 − 1.06i)6-s + (0.930 + 0.622i)7-s + (0.923 − 0.382i)8-s + (−0.596 − 0.247i)9-s + (−1.86 + 2.79i)11-s + (1.58 − 1.06i)12-s + 1.47·13-s + (−0.930 + 0.622i)14-s + i·16-s + (−3.90 + 1.32i)17-s + (0.456 − 0.456i)18-s + (−4.07 + 1.68i)19-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.215 + 1.08i)3-s + (−0.353 − 0.353i)4-s + (−0.648 − 0.433i)6-s + (0.351 + 0.235i)7-s + (0.326 − 0.135i)8-s + (−0.198 − 0.0823i)9-s + (−0.562 + 0.842i)11-s + (0.458 − 0.306i)12-s + 0.409·13-s + (−0.248 + 0.166i)14-s + 0.250i·16-s + (−0.946 + 0.322i)17-s + (0.107 − 0.107i)18-s + (−0.934 + 0.386i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((−0.918+0.396i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((−0.918+0.396i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: −0.918+0.396i-0.918 + 0.396i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(607,⋅)\chi_{850} (607, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), −0.918+0.396i)(2,\ 850,\ (\ :1/2),\ -0.918 + 0.396i)

Particular Values

L(1)L(1) ≈\approx 0.171611−0.830378i0.171611 - 0.830378i
L(12)L(\frac12) ≈\approx 0.171611−0.830378i0.171611 - 0.830378i
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)
bad2 1+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
5 1 1
17 1+(3.90−1.32i)T 1 + (3.90 - 1.32i)T
good3 1+(0.372−1.87i)T+(−2.77−1.14i)T2 1 + (0.372 - 1.87i)T + (-2.77 - 1.14i)T^{2}
7 1+(−0.930−0.622i)T+(2.67+6.46i)T2 1 + (-0.930 - 0.622i)T + (2.67 + 6.46i)T^{2}
11 1+(1.86−2.79i)T+(−4.20−10.1i)T2 1 + (1.86 - 2.79i)T + (-4.20 - 10.1i)T^{2}
13 1−1.47T+13T2 1 - 1.47T + 13T^{2}
19 1+(4.07−1.68i)T+(13.4−13.4i)T2 1 + (4.07 - 1.68i)T + (13.4 - 13.4i)T^{2}
23 1+(−2.92+0.581i)T+(21.2−8.80i)T2 1 + (-2.92 + 0.581i)T + (21.2 - 8.80i)T^{2}
29 1+(−3.34−0.664i)T+(26.7+11.0i)T2 1 + (-3.34 - 0.664i)T + (26.7 + 11.0i)T^{2}
31 1+(−2.94−4.41i)T+(−11.8+28.6i)T2 1 + (-2.94 - 4.41i)T + (-11.8 + 28.6i)T^{2}
37 1+(8.28+1.64i)T+(34.1+14.1i)T2 1 + (8.28 + 1.64i)T + (34.1 + 14.1i)T^{2}
41 1+(10.6−2.11i)T+(37.8−15.6i)T2 1 + (10.6 - 2.11i)T + (37.8 - 15.6i)T^{2}
43 1+(2.55+6.16i)T+(−30.4+30.4i)T2 1 + (2.55 + 6.16i)T + (-30.4 + 30.4i)T^{2}
47 1−6.49iT−47T2 1 - 6.49iT - 47T^{2}
53 1+(−5.24−2.17i)T+(37.4+37.4i)T2 1 + (-5.24 - 2.17i)T + (37.4 + 37.4i)T^{2}
59 1+(−0.784+1.89i)T+(−41.7−41.7i)T2 1 + (-0.784 + 1.89i)T + (-41.7 - 41.7i)T^{2}
61 1+(−0.266−1.34i)T+(−56.3+23.3i)T2 1 + (-0.266 - 1.34i)T + (-56.3 + 23.3i)T^{2}
67 1+(−7.14+7.14i)T−67iT2 1 + (-7.14 + 7.14i)T - 67iT^{2}
71 1+(−1.30+0.874i)T+(27.1−65.5i)T2 1 + (-1.30 + 0.874i)T + (27.1 - 65.5i)T^{2}
73 1+(−10.5+7.04i)T+(27.9−67.4i)T2 1 + (-10.5 + 7.04i)T + (27.9 - 67.4i)T^{2}
79 1+(6.00+4.01i)T+(30.2+72.9i)T2 1 + (6.00 + 4.01i)T + (30.2 + 72.9i)T^{2}
83 1+(4.89−11.8i)T+(−58.6−58.6i)T2 1 + (4.89 - 11.8i)T + (-58.6 - 58.6i)T^{2}
89 1+(5.20+5.20i)T+89iT2 1 + (5.20 + 5.20i)T + 89iT^{2}
97 1+(−8.14+5.44i)T+(37.1−89.6i)T2 1 + (-8.14 + 5.44i)T + (37.1 - 89.6i)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

−10.43917689779655176463884475097, −9.942662735310211758226382441430, −8.823233243804582810106721422184, −8.395034685197999485264642326131, −7.16774016508325371782183417107, −6.38857050764620712037757894607, −5.11854059658977747353921019134, −4.73674732517538653453143547820, −3.64883972648396767820110836577, −1.93536642243602029178715292055, 0.46179083225507098873941451878, 1.72119195383946792669663530080, 2.81986309136565615235299307770, 4.13644507921078765424932124404, 5.25386610596044527633538069625, 6.48450445400904076137782544217, 7.08538040358457892148401942445, 8.275239222781451963664594688728, 8.573793926897503081026415027171, 9.848806232209406997357720888474

Graph of the ZZ-function along the critical line