↑

Properties

Label 2-21e2-147.101-c1-0-11
Degree 22
Conductor 441441
Sign −0.982−0.183i-0.982 - 0.183i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  + (−1.72 − 0.677i)2-s + (1.05 + 0.975i)4-s + (−0.820 − 0.559i)5-s + (0.202 + 2.63i)7-s + (0.454 + 0.944i)8-s + (1.03 + 1.52i)10-s + (−0.669 − 4.44i)11-s + (0.491 − 0.391i)13-s + (1.43 − 4.68i)14-s + (−0.359 − 4.79i)16-s + (−7.67 − 2.36i)17-s + (0.358 − 0.206i)19-s + (−0.316 − 1.38i)20-s + (−1.85 + 8.11i)22-s + (2.22 + 7.21i)23-s + ⋯
L(s)  = 1  + (−1.21 − 0.478i)2-s + (0.525 + 0.487i)4-s + (−0.366 − 0.250i)5-s + (0.0763 + 0.997i)7-s + (0.160 + 0.333i)8-s + (0.327 + 0.480i)10-s + (−0.201 − 1.33i)11-s + (0.136 − 0.108i)13-s + (0.384 − 1.25i)14-s + (−0.0898 − 1.19i)16-s + (−1.86 − 0.574i)17-s + (0.0821 − 0.0474i)19-s + (−0.0708 − 0.310i)20-s + (−0.394 + 1.73i)22-s + (0.463 + 1.50i)23-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.982−0.183i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((−0.982−0.183i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.982−0.183i-0.982 - 0.183i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(395,⋅)\chi_{441} (395, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), −0.982−0.183i)(2,\ 441,\ (\ :1/2),\ -0.982 - 0.183i)

Particular Values

L(1)L(1) ≈\approx 0.0121992+0.131653i0.0121992 + 0.131653i
L(12)L(\frac12) ≈\approx 0.0121992+0.131653i0.0121992 + 0.131653i
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)
bad3 1 1
7 1+(−0.202−2.63i)T 1 + (-0.202 - 2.63i)T
good2 1+(1.72+0.677i)T+(1.46+1.36i)T2 1 + (1.72 + 0.677i)T + (1.46 + 1.36i)T^{2}
5 1+(0.820+0.559i)T+(1.82+4.65i)T2 1 + (0.820 + 0.559i)T + (1.82 + 4.65i)T^{2}
11 1+(0.669+4.44i)T+(−10.5+3.24i)T2 1 + (0.669 + 4.44i)T + (-10.5 + 3.24i)T^{2}
13 1+(−0.491+0.391i)T+(2.89−12.6i)T2 1 + (-0.491 + 0.391i)T + (2.89 - 12.6i)T^{2}
17 1+(7.67+2.36i)T+(14.0+9.57i)T2 1 + (7.67 + 2.36i)T + (14.0 + 9.57i)T^{2}
19 1+(−0.358+0.206i)T+(9.5−16.4i)T2 1 + (-0.358 + 0.206i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.22−7.21i)T+(−19.0+12.9i)T2 1 + (-2.22 - 7.21i)T + (-19.0 + 12.9i)T^{2}
29 1+(−0.523+0.119i)T+(26.1−12.5i)T2 1 + (-0.523 + 0.119i)T + (26.1 - 12.5i)T^{2}
31 1+(7.47+4.31i)T+(15.5+26.8i)T2 1 + (7.47 + 4.31i)T + (15.5 + 26.8i)T^{2}
37 1+(5.17−4.79i)T+(2.76−36.8i)T2 1 + (5.17 - 4.79i)T + (2.76 - 36.8i)T^{2}
41 1+(4.20−2.02i)T+(25.5−32.0i)T2 1 + (4.20 - 2.02i)T + (25.5 - 32.0i)T^{2}
43 1+(6.50+3.13i)T+(26.8+33.6i)T2 1 + (6.50 + 3.13i)T + (26.8 + 33.6i)T^{2}
47 1+(−2.32+5.92i)T+(−34.4−31.9i)T2 1 + (-2.32 + 5.92i)T + (-34.4 - 31.9i)T^{2}
53 1+(1.40−1.51i)T+(−3.96−52.8i)T2 1 + (1.40 - 1.51i)T + (-3.96 - 52.8i)T^{2}
59 1+(10.2−6.98i)T+(21.5−54.9i)T2 1 + (10.2 - 6.98i)T + (21.5 - 54.9i)T^{2}
61 1+(−3.28−3.53i)T+(−4.55+60.8i)T2 1 + (-3.28 - 3.53i)T + (-4.55 + 60.8i)T^{2}
67 1+(−1.75+3.03i)T+(−33.5−58.0i)T2 1 + (-1.75 + 3.03i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.14−0.488i)T+(63.9+30.8i)T2 1 + (-2.14 - 0.488i)T + (63.9 + 30.8i)T^{2}
73 1+(−2.40+0.944i)T+(53.5−49.6i)T2 1 + (-2.40 + 0.944i)T + (53.5 - 49.6i)T^{2}
79 1+(6.85+11.8i)T+(−39.5+68.4i)T2 1 + (6.85 + 11.8i)T + (-39.5 + 68.4i)T^{2}
83 1+(−6.36+7.98i)T+(−18.4−80.9i)T2 1 + (-6.36 + 7.98i)T + (-18.4 - 80.9i)T^{2}
89 1+(6.09+0.918i)T+(85.0+26.2i)T2 1 + (6.09 + 0.918i)T + (85.0 + 26.2i)T^{2}
97 1+12.9iT−97T2 1 + 12.9iT - 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

−10.70146025750958625549283692101, −9.541044247890682092412729061550, −8.769900716762302066950266547180, −8.419583489011863423002726273955, −7.27022371449380771755221618026, −5.90078982340978292657216410645, −4.89572349644597279093651270102, −3.20395348988360544351836548425, −1.92978071778703070801923235688, −0.12032281567365684721157881906, 1.81473220747313354341853401198, 3.83569147160427350005410994147, 4.75112077650554837986533349982, 6.74057849630825829711197263278, 6.97522822187680867943563350358, 7.959287419123358532259025372401, 8.837838725647246162982332602305, 9.680399345055092292302688468890, 10.67588882277450267344921875630, 10.99405039485617480284896759952

Graph of the ZZ-function along the critical line