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Properties

Label 2-850-85.82-c1-0-25
Degree 22
Conductor 850850
Sign −0.509+0.860i-0.509 + 0.860i
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  + (0.923 + 0.382i)2-s + (−0.530 − 0.793i)3-s + (0.707 + 0.707i)4-s + (−0.186 − 0.935i)6-s + (−0.736 − 3.70i)7-s + (0.382 + 0.923i)8-s + (0.799 − 1.93i)9-s + (−2.74 + 0.546i)11-s + (0.186 − 0.935i)12-s − 5.74·13-s + (0.736 − 3.70i)14-s + i·16-s + (−3.33 − 2.41i)17-s + (1.47 − 1.47i)18-s + (0.119 + 0.288i)19-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (−0.306 − 0.458i)3-s + (0.353 + 0.353i)4-s + (−0.0760 − 0.382i)6-s + (−0.278 − 1.39i)7-s + (0.135 + 0.326i)8-s + (0.266 − 0.643i)9-s + (−0.828 + 0.164i)11-s + (0.0537 − 0.270i)12-s − 1.59·13-s + (0.196 − 0.989i)14-s + 0.250i·16-s + (−0.809 − 0.586i)17-s + (0.348 − 0.348i)18-s + (0.0273 + 0.0660i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((−0.509+0.860i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((−0.509+0.860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: −0.509+0.860i-0.509 + 0.860i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(507,⋅)\chi_{850} (507, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), −0.509+0.860i)(2,\ 850,\ (\ :1/2),\ -0.509 + 0.860i)

Particular Values

L(1)L(1) ≈\approx 0.610974−1.07243i0.610974 - 1.07243i
L(12)L(\frac12) ≈\approx 0.610974−1.07243i0.610974 - 1.07243i
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+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
5 1 1
17 1+(3.33+2.41i)T 1 + (3.33 + 2.41i)T
good3 1+(0.530+0.793i)T+(−1.14+2.77i)T2 1 + (0.530 + 0.793i)T + (-1.14 + 2.77i)T^{2}
7 1+(0.736+3.70i)T+(−6.46+2.67i)T2 1 + (0.736 + 3.70i)T + (-6.46 + 2.67i)T^{2}
11 1+(2.74−0.546i)T+(10.1−4.20i)T2 1 + (2.74 - 0.546i)T + (10.1 - 4.20i)T^{2}
13 1+5.74T+13T2 1 + 5.74T + 13T^{2}
19 1+(−0.119−0.288i)T+(−13.4+13.4i)T2 1 + (-0.119 - 0.288i)T + (-13.4 + 13.4i)T^{2}
23 1+(−3.97−2.65i)T+(8.80+21.2i)T2 1 + (-3.97 - 2.65i)T + (8.80 + 21.2i)T^{2}
29 1+(−0.463+0.309i)T+(11.0−26.7i)T2 1 + (-0.463 + 0.309i)T + (11.0 - 26.7i)T^{2}
31 1+(5.00+0.994i)T+(28.6+11.8i)T2 1 + (5.00 + 0.994i)T + (28.6 + 11.8i)T^{2}
37 1+(−8.56+5.72i)T+(14.1−34.1i)T2 1 + (-8.56 + 5.72i)T + (14.1 - 34.1i)T^{2}
41 1+(7.70+5.15i)T+(15.6+37.8i)T2 1 + (7.70 + 5.15i)T + (15.6 + 37.8i)T^{2}
43 1+(−8.32+3.44i)T+(30.4−30.4i)T2 1 + (-8.32 + 3.44i)T + (30.4 - 30.4i)T^{2}
47 1−5.61iT−47T2 1 - 5.61iT - 47T^{2}
53 1+(−2.52+6.08i)T+(−37.4−37.4i)T2 1 + (-2.52 + 6.08i)T + (-37.4 - 37.4i)T^{2}
59 1+(−3.14−1.30i)T+(41.7+41.7i)T2 1 + (-3.14 - 1.30i)T + (41.7 + 41.7i)T^{2}
61 1+(−5.67+8.48i)T+(−23.3−56.3i)T2 1 + (-5.67 + 8.48i)T + (-23.3 - 56.3i)T^{2}
67 1+(1.94−1.94i)T−67iT2 1 + (1.94 - 1.94i)T - 67iT^{2}
71 1+(−0.782+3.93i)T+(−65.5−27.1i)T2 1 + (-0.782 + 3.93i)T + (-65.5 - 27.1i)T^{2}
73 1+(−0.167+0.841i)T+(−67.4−27.9i)T2 1 + (-0.167 + 0.841i)T + (-67.4 - 27.9i)T^{2}
79 1+(0.872+4.38i)T+(−72.9+30.2i)T2 1 + (0.872 + 4.38i)T + (-72.9 + 30.2i)T^{2}
83 1+(−1.36−0.565i)T+(58.6+58.6i)T2 1 + (-1.36 - 0.565i)T + (58.6 + 58.6i)T^{2}
89 1+(−7.49−7.49i)T+89iT2 1 + (-7.49 - 7.49i)T + 89iT^{2}
97 1+(−3.04+15.3i)T+(−89.6−37.1i)T2 1 + (-3.04 + 15.3i)T + (-89.6 - 37.1i)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

−9.965692575567935888145506754377, −9.187391581330768265404139667874, −7.55504975086112087683345611955, −7.33913118774258279613577546589, −6.61754552371347711656516196507, −5.42299641530745659074735238721, −4.55903818714632528851569676965, −3.60248807865581712764979721009, −2.31211877845741735807875319646, −0.46344320223444250675836865542, 2.22612412342876828070418106358, 2.83590558810903969802851839778, 4.43347630457304102977826770628, 5.10413925523247121901106264235, 5.76555029380789679621879622492, 6.86276155048658872241105772435, 7.906480734458418218782270047463, 8.941471852678795167157901073615, 9.823749792457605927145128987725, 10.52118788088564593958563169982

Graph of the ZZ-function along the critical line