↑

Properties

Label 2-84-12.11-c5-0-23
Degree 22
Conductor 8484
Sign 0.979−0.200i0.979 - 0.200i
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.66 − 3.20i)2-s + (15.3 + 2.55i)3-s + (11.4 + 29.8i)4-s − 8.37i·5-s + (−63.5 − 61.1i)6-s + 49i·7-s + (42.0 − 176. i)8-s + (229. + 78.7i)9-s + (−26.8 + 39.0i)10-s + 453.·11-s + (100. + 488. i)12-s − 689.·13-s + (156. − 228. i)14-s + (21.4 − 128. i)15-s + (−759. + 686. i)16-s + 2.11e3i·17-s + ⋯
L(s)  = 1  + (−0.824 − 0.566i)2-s + (0.986 + 0.164i)3-s + (0.359 + 0.933i)4-s − 0.149i·5-s + (−0.720 − 0.693i)6-s + 0.377i·7-s + (0.232 − 0.972i)8-s + (0.946 + 0.323i)9-s + (−0.0848 + 0.123i)10-s + 1.12·11-s + (0.200 + 0.979i)12-s − 1.13·13-s + (0.213 − 0.311i)14-s + (0.0245 − 0.147i)15-s + (−0.742 + 0.670i)16-s + 1.77i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.979−0.200i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((0.979−0.200i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.979−0.200i0.979 - 0.200i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(71,⋅)\chi_{84} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), 0.979−0.200i)(2,\ 84,\ (\ :5/2),\ 0.979 - 0.200i)

Particular Values

L(3)L(3) ≈\approx 1.76437+0.179107i1.76437 + 0.179107i
L(12)L(\frac12) ≈\approx 1.76437+0.179107i1.76437 + 0.179107i
L(72)L(\frac{7}{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+(4.66+3.20i)T 1 + (4.66 + 3.20i)T
3 1+(−15.3−2.55i)T 1 + (-15.3 - 2.55i)T
7 1−49iT 1 - 49iT
good5 1+8.37iT−3.12e3T2 1 + 8.37iT - 3.12e3T^{2}
11 1−453.T+1.61e5T2 1 - 453.T + 1.61e5T^{2}
13 1+689.T+3.71e5T2 1 + 689.T + 3.71e5T^{2}
17 1−2.11e3iT−1.41e6T2 1 - 2.11e3iT - 1.41e6T^{2}
19 1−129.iT−2.47e6T2 1 - 129. iT - 2.47e6T^{2}
23 1−4.27e3T+6.43e6T2 1 - 4.27e3T + 6.43e6T^{2}
29 1+4.93e3iT−2.05e7T2 1 + 4.93e3iT - 2.05e7T^{2}
31 1−1.62e3iT−2.86e7T2 1 - 1.62e3iT - 2.86e7T^{2}
37 1−4.59e3T+6.93e7T2 1 - 4.59e3T + 6.93e7T^{2}
41 1−675.iT−1.15e8T2 1 - 675. iT - 1.15e8T^{2}
43 1−1.83e4iT−1.47e8T2 1 - 1.83e4iT - 1.47e8T^{2}
47 1−1.04e4T+2.29e8T2 1 - 1.04e4T + 2.29e8T^{2}
53 1+2.33e4iT−4.18e8T2 1 + 2.33e4iT - 4.18e8T^{2}
59 1+2.36e4T+7.14e8T2 1 + 2.36e4T + 7.14e8T^{2}
61 1+2.63e4T+8.44e8T2 1 + 2.63e4T + 8.44e8T^{2}
67 1+1.64e4iT−1.35e9T2 1 + 1.64e4iT - 1.35e9T^{2}
71 1+1.11e4T+1.80e9T2 1 + 1.11e4T + 1.80e9T^{2}
73 1−1.83e4T+2.07e9T2 1 - 1.83e4T + 2.07e9T^{2}
79 1−8.46e4iT−3.07e9T2 1 - 8.46e4iT - 3.07e9T^{2}
83 1+4.43e4T+3.93e9T2 1 + 4.43e4T + 3.93e9T^{2}
89 1+1.17e5iT−5.58e9T2 1 + 1.17e5iT - 5.58e9T^{2}
97 1+1.13e5T+8.58e9T2 1 + 1.13e5T + 8.58e9T^{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

−13.03572055254735068427078128026, −12.31640178588476648075762148954, −10.95105856910606139309060297950, −9.750022604619014689021021493792, −8.977411401599375123713806211162, −8.029024362215871895070243771087, −6.74519603155264523005466903081, −4.30995640228323175718390032149, −2.89875800665064296402284405232, −1.46491708277735980410321723261, 1.00694290792320026213412330618, 2.77379554800675668262170541420, 4.83035712695565756541298253118, 6.93409937736225467967511657302, 7.34161834299221166708665939713, 8.961065668812671263022791802185, 9.444226465559637482023956185259, 10.76341789306733743254625194084, 12.14796457179370208429386354071, 13.70862067856135357742011737220

Graph of the ZZ-function along the critical line