↑

Properties

Label 2-1176-8.5-c1-0-62
Degree 22
Conductor 11761176
Sign 0.570+0.821i0.570 + 0.821i
Analytic cond. 9.390409.39040
Root an. cond. 3.064373.06437
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  + (1.38 − 0.284i)2-s − i·3-s + (1.83 − 0.788i)4-s + 2.29i·5-s + (−0.284 − 1.38i)6-s + (2.32 − 1.61i)8-s − 9-s + (0.652 + 3.17i)10-s − 3.88i·11-s + (−0.788 − 1.83i)12-s − 3.33i·13-s + 2.29·15-s + (2.75 − 2.89i)16-s + 0.287·17-s + (−1.38 + 0.284i)18-s + 2.78i·19-s + ⋯
L(s)  = 1  + (0.979 − 0.201i)2-s − 0.577i·3-s + (0.919 − 0.394i)4-s + 1.02i·5-s + (−0.116 − 0.565i)6-s + (0.821 − 0.570i)8-s − 0.333·9-s + (0.206 + 1.00i)10-s − 1.17i·11-s + (−0.227 − 0.530i)12-s − 0.924i·13-s + 0.592·15-s + (0.689 − 0.724i)16-s + 0.0696·17-s + (−0.326 + 0.0670i)18-s + 0.638i·19-s + ⋯

Functional equation

Λ(s)=(1176s/2 ΓC(s) L(s)=((0.570+0.821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1176s/2 ΓC(s+1/2) L(s)=((0.570+0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11761176    =    23⋅3⋅722^{3} \cdot 3 \cdot 7^{2}
Sign: 0.570+0.821i0.570 + 0.821i
Analytic conductor: 9.390409.39040
Root analytic conductor: 3.064373.06437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1176(589,⋅)\chi_{1176} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1176, ( :1/2), 0.570+0.821i)(2,\ 1176,\ (\ :1/2),\ 0.570 + 0.821i)

Particular Values

L(1)L(1) ≈\approx 3.1591131993.159113199
L(12)L(\frac12) ≈\approx 3.1591131993.159113199
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+(−1.38+0.284i)T 1 + (-1.38 + 0.284i)T
3 1+iT 1 + iT
7 1 1
good5 1−2.29iT−5T2 1 - 2.29iT - 5T^{2}
11 1+3.88iT−11T2 1 + 3.88iT - 11T^{2}
13 1+3.33iT−13T2 1 + 3.33iT - 13T^{2}
17 1−0.287T+17T2 1 - 0.287T + 17T^{2}
19 1−2.78iT−19T2 1 - 2.78iT - 19T^{2}
23 1−6.53T+23T2 1 - 6.53T + 23T^{2}
29 1+5.53iT−29T2 1 + 5.53iT - 29T^{2}
31 1−7.44T+31T2 1 - 7.44T + 31T^{2}
37 1+5.95iT−37T2 1 + 5.95iT - 37T^{2}
41 1+3.51T+41T2 1 + 3.51T + 41T^{2}
43 1−11.2iT−43T2 1 - 11.2iT - 43T^{2}
47 1−0.0870T+47T2 1 - 0.0870T + 47T^{2}
53 1−7.05iT−53T2 1 - 7.05iT - 53T^{2}
59 1−4.35iT−59T2 1 - 4.35iT - 59T^{2}
61 1−7.16iT−61T2 1 - 7.16iT - 61T^{2}
67 1−13.0iT−67T2 1 - 13.0iT - 67T^{2}
71 1+6.18T+71T2 1 + 6.18T + 71T^{2}
73 1+13.8T+73T2 1 + 13.8T + 73T^{2}
79 1+8.99T+79T2 1 + 8.99T + 79T^{2}
83 1+17.6iT−83T2 1 + 17.6iT - 83T^{2}
89 1+17.1T+89T2 1 + 17.1T + 89T^{2}
97 1−6.46T+97T2 1 - 6.46T + 97T^{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

−10.05731180533401289451572450174, −8.653725498413360988664545888028, −7.71536665416778399533912934490, −7.02727572644110572327724388268, −6.08373128480821660270033953207, −5.69059851095762075778364390229, −4.35617049502888137394547896143, −3.03866280580582377963606065555, −2.82640320801382722441521484157, −1.12084594579307544412662433524, 1.58221013512802512833152053653, 2.88760348580247676857931270739, 4.08331482637001121531241446654, 4.89743067841845845555322476567, 5.12911828299670014468759282368, 6.58652243562235685294951630444, 7.12684501103480246191171871086, 8.347947238876001265953811879296, 9.021936369990172342351499164659, 9.896654557714500448986387941284

Graph of the ZZ-function along the critical line