↑

Properties

Label 2-850-85.23-c1-0-5
Degree 22
Conductor 850850
Sign 0.637−0.770i0.637 - 0.770i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)2-s + (−1.41 − 0.942i)3-s + (0.707 − 0.707i)4-s + (1.66 + 0.330i)6-s + (5.02 + 0.999i)7-s + (−0.382 + 0.923i)8-s + (−0.0471 − 0.113i)9-s + (−0.999 + 5.02i)11-s + (−1.66 + 0.330i)12-s + 3.13·13-s + (−5.02 + 0.999i)14-s − i·16-s + (−3.59 + 2.02i)17-s + (0.0871 + 0.0871i)18-s + (−0.297 + 0.718i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.814 − 0.544i)3-s + (0.353 − 0.353i)4-s + (0.679 + 0.135i)6-s + (1.89 + 0.377i)7-s + (−0.135 + 0.326i)8-s + (−0.0157 − 0.0379i)9-s + (−0.301 + 1.51i)11-s + (−0.480 + 0.0955i)12-s + 0.870·13-s + (−1.34 + 0.267i)14-s − 0.250i·16-s + (−0.871 + 0.490i)17-s + (0.0205 + 0.0205i)18-s + (−0.0682 + 0.164i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((0.637−0.770i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((0.637−0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: 0.637−0.770i0.637 - 0.770i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(193,⋅)\chi_{850} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), 0.637−0.770i)(2,\ 850,\ (\ :1/2),\ 0.637 - 0.770i)

Particular Values

L(1)L(1) ≈\approx 0.863166+0.406084i0.863166 + 0.406084i
L(12)L(\frac12) ≈\approx 0.863166+0.406084i0.863166 + 0.406084i
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+(3.59−2.02i)T 1 + (3.59 - 2.02i)T
good3 1+(1.41+0.942i)T+(1.14+2.77i)T2 1 + (1.41 + 0.942i)T + (1.14 + 2.77i)T^{2}
7 1+(−5.02−0.999i)T+(6.46+2.67i)T2 1 + (-5.02 - 0.999i)T + (6.46 + 2.67i)T^{2}
11 1+(0.999−5.02i)T+(−10.1−4.20i)T2 1 + (0.999 - 5.02i)T + (-10.1 - 4.20i)T^{2}
13 1−3.13T+13T2 1 - 3.13T + 13T^{2}
19 1+(0.297−0.718i)T+(−13.4−13.4i)T2 1 + (0.297 - 0.718i)T + (-13.4 - 13.4i)T^{2}
23 1+(0.327+0.489i)T+(−8.80+21.2i)T2 1 + (0.327 + 0.489i)T + (-8.80 + 21.2i)T^{2}
29 1+(1.78−2.66i)T+(−11.0−26.7i)T2 1 + (1.78 - 2.66i)T + (-11.0 - 26.7i)T^{2}
31 1+(0.00940+0.0472i)T+(−28.6+11.8i)T2 1 + (0.00940 + 0.0472i)T + (-28.6 + 11.8i)T^{2}
37 1+(−1.22+1.83i)T+(−14.1−34.1i)T2 1 + (-1.22 + 1.83i)T + (-14.1 - 34.1i)T^{2}
41 1+(−0.774−1.15i)T+(−15.6+37.8i)T2 1 + (-0.774 - 1.15i)T + (-15.6 + 37.8i)T^{2}
43 1+(2.55+1.05i)T+(30.4+30.4i)T2 1 + (2.55 + 1.05i)T + (30.4 + 30.4i)T^{2}
47 1−1.90iT−47T2 1 - 1.90iT - 47T^{2}
53 1+(−4.43−10.6i)T+(−37.4+37.4i)T2 1 + (-4.43 - 10.6i)T + (-37.4 + 37.4i)T^{2}
59 1+(−2.75+1.14i)T+(41.7−41.7i)T2 1 + (-2.75 + 1.14i)T + (41.7 - 41.7i)T^{2}
61 1+(8.20−5.48i)T+(23.3−56.3i)T2 1 + (8.20 - 5.48i)T + (23.3 - 56.3i)T^{2}
67 1+(−7.07−7.07i)T+67iT2 1 + (-7.07 - 7.07i)T + 67iT^{2}
71 1+(−7.55+1.50i)T+(65.5−27.1i)T2 1 + (-7.55 + 1.50i)T + (65.5 - 27.1i)T^{2}
73 1+(7.03−1.39i)T+(67.4−27.9i)T2 1 + (7.03 - 1.39i)T + (67.4 - 27.9i)T^{2}
79 1+(−9.46−1.88i)T+(72.9+30.2i)T2 1 + (-9.46 - 1.88i)T + (72.9 + 30.2i)T^{2}
83 1+(−15.6+6.48i)T+(58.6−58.6i)T2 1 + (-15.6 + 6.48i)T + (58.6 - 58.6i)T^{2}
89 1+(−5.07+5.07i)T−89iT2 1 + (-5.07 + 5.07i)T - 89iT^{2}
97 1+(1.77−0.353i)T+(89.6−37.1i)T2 1 + (1.77 - 0.353i)T + (89.6 - 37.1i)T^{2}
show more
show less
   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

−10.60121926053717449205217242800, −9.275996157774977522275234134958, −8.528721685681358941438101621186, −7.71989102152843851387737312590, −6.99528342544667364460477575504, −6.02096759732494127148846282996, −5.18988598613300056556348426949, −4.29410555821626699760728847639, −2.13500671523746053300051050288, −1.33151420697606896818492962723, 0.71803945126421930112274718240, 2.14258752055984169883396487528, 3.74072341682250584046313996626, 4.80272419705213370071599262041, 5.53314417873792835756086463903, 6.57633861813023007818487222384, 7.954722145459064834193418693674, 8.258629397643026973100692061127, 9.184332951834521324475775178483, 10.46310833408983131513373863805

Graph of the ZZ-function along the critical line