↑

Properties

Label 2-85-17.9-c1-0-2
Degree 22
Conductor 8585
Sign 0.789+0.614i0.789 + 0.614i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.93i)2-s + (−1.42 − 0.591i)3-s − 5.45i·4-s + (0.382 − 0.923i)5-s + (3.90 − 1.61i)6-s + (−0.483 − 1.16i)7-s + (6.67 + 6.67i)8-s + (−0.429 − 0.429i)9-s + (1.04 + 2.52i)10-s + (0.386 − 0.159i)11-s + (−3.22 + 7.79i)12-s − 5.66i·13-s + (3.18 + 1.32i)14-s + (−1.09 + 1.09i)15-s − 14.8·16-s + (1.27 − 3.92i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (−0.825 − 0.341i)3-s − 2.72i·4-s + (0.171 − 0.413i)5-s + (1.59 − 0.659i)6-s + (−0.182 − 0.441i)7-s + (2.35 + 2.35i)8-s + (−0.143 − 0.143i)9-s + (0.330 + 0.797i)10-s + (0.116 − 0.0482i)11-s + (−0.932 + 2.25i)12-s − 1.57i·13-s + (0.852 + 0.353i)14-s + (−0.282 + 0.282i)15-s − 3.71·16-s + (0.309 − 0.950i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.789+0.614i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.789+0.614i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.789+0.614i0.789 + 0.614i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(26,⋅)\chi_{85} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.789+0.614i)(2,\ 85,\ (\ :1/2),\ 0.789 + 0.614i)

Particular Values

L(1)L(1) ≈\approx 0.305878−0.104999i0.305878 - 0.104999i
L(12)L(\frac12) ≈\approx 0.305878−0.104999i0.305878 - 0.104999i
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+(−0.382+0.923i)T 1 + (-0.382 + 0.923i)T
17 1+(−1.27+3.92i)T 1 + (-1.27 + 3.92i)T
good2 1+(1.93−1.93i)T−2iT2 1 + (1.93 - 1.93i)T - 2iT^{2}
3 1+(1.42+0.591i)T+(2.12+2.12i)T2 1 + (1.42 + 0.591i)T + (2.12 + 2.12i)T^{2}
7 1+(0.483+1.16i)T+(−4.94+4.94i)T2 1 + (0.483 + 1.16i)T + (-4.94 + 4.94i)T^{2}
11 1+(−0.386+0.159i)T+(7.77−7.77i)T2 1 + (-0.386 + 0.159i)T + (7.77 - 7.77i)T^{2}
13 1+5.66iT−13T2 1 + 5.66iT - 13T^{2}
19 1+(0.0948−0.0948i)T−19iT2 1 + (0.0948 - 0.0948i)T - 19iT^{2}
23 1+(6.30−2.60i)T+(16.2−16.2i)T2 1 + (6.30 - 2.60i)T + (16.2 - 16.2i)T^{2}
29 1+(0.126−0.304i)T+(−20.5−20.5i)T2 1 + (0.126 - 0.304i)T + (-20.5 - 20.5i)T^{2}
31 1+(0.559+0.231i)T+(21.9+21.9i)T2 1 + (0.559 + 0.231i)T + (21.9 + 21.9i)T^{2}
37 1+(−8.16−3.38i)T+(26.1+26.1i)T2 1 + (-8.16 - 3.38i)T + (26.1 + 26.1i)T^{2}
41 1+(−0.625−1.50i)T+(−28.9+28.9i)T2 1 + (-0.625 - 1.50i)T + (-28.9 + 28.9i)T^{2}
43 1+(1.41+1.41i)T+43iT2 1 + (1.41 + 1.41i)T + 43iT^{2}
47 1+5.06iT−47T2 1 + 5.06iT - 47T^{2}
53 1+(1.09−1.09i)T−53iT2 1 + (1.09 - 1.09i)T - 53iT^{2}
59 1+(−0.997−0.997i)T+59iT2 1 + (-0.997 - 0.997i)T + 59iT^{2}
61 1+(−2.98−7.21i)T+(−43.1+43.1i)T2 1 + (-2.98 - 7.21i)T + (-43.1 + 43.1i)T^{2}
67 1−12.1T+67T2 1 - 12.1T + 67T^{2}
71 1+(4.45+1.84i)T+(50.2+50.2i)T2 1 + (4.45 + 1.84i)T + (50.2 + 50.2i)T^{2}
73 1+(1.10−2.67i)T+(−51.6−51.6i)T2 1 + (1.10 - 2.67i)T + (-51.6 - 51.6i)T^{2}
79 1+(−4.00+1.65i)T+(55.8−55.8i)T2 1 + (-4.00 + 1.65i)T + (55.8 - 55.8i)T^{2}
83 1+(−3.10+3.10i)T−83iT2 1 + (-3.10 + 3.10i)T - 83iT^{2}
89 1+4.98iT−89T2 1 + 4.98iT - 89T^{2}
97 1+(−0.243+0.587i)T+(−68.5−68.5i)T2 1 + (-0.243 + 0.587i)T + (-68.5 - 68.5i)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

−14.51266238428336667858855354922, −13.30032005719569766266001815068, −11.71157898823074063664904482539, −10.44048439188333063627250073768, −9.589131899389589947362526283906, −8.287211704614109394575445016164, −7.30918999068698331973787007628, −6.09605228828348968395098163212, −5.30792889446107921951414610202, −0.69569041440203107906335757608, 2.21195337879718829362405162580, 4.09793045906188679932659025571, 6.36791281353459152180680864396, 8.001759764439202344084297525463, 9.207679015789890461103200196241, 10.12672219570034267159152919679, 11.05458141973646286076921162224, 11.74897272786230571670296904970, 12.63488691906418355366101202956, 14.14936269376982037164473616966

Graph of the ZZ-function along the critical line