↑

Properties

Label 2-1620-9.4-c3-0-33
Degree 22
Conductor 16201620
Sign 0.939+0.342i0.939 + 0.342i
Analytic cond. 95.583095.5830
Root an. cond. 9.776669.77666
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (14 + 24.2i)7-s + (12 + 20.7i)11-s + (35 − 60.6i)13-s + 102·17-s + 20·19-s + (36 − 62.3i)23-s + (−12.5 − 21.6i)25-s + (−153 − 265. i)29-s + (68 − 117. i)31-s + 140·35-s − 214·37-s + (75 − 129. i)41-s + (146 + 252. i)43-s + (36 + 62.3i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (0.328 + 0.569i)11-s + (0.746 − 1.29i)13-s + 1.45·17-s + 0.241·19-s + (0.326 − 0.565i)23-s + (−0.100 − 0.173i)25-s + (−0.979 − 1.69i)29-s + (0.393 − 0.682i)31-s + 0.676·35-s − 0.950·37-s + (0.285 − 0.494i)41-s + (0.517 + 0.896i)43-s + (0.111 + 0.193i)47-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+3/2) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: 0.939+0.342i0.939 + 0.342i
Analytic conductor: 95.583095.5830
Root analytic conductor: 9.776669.77666
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1620(1081,⋅)\chi_{1620} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :3/2), 0.939+0.342i)(2,\ 1620,\ (\ :3/2),\ 0.939 + 0.342i)

Particular Values

L(2)L(2) ≈\approx 2.9252889772.925288977
L(12)L(\frac12) ≈\approx 2.9252889772.925288977
L(52)L(\frac{5}{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
3 1 1
5 1+(−2.5+4.33i)T 1 + (-2.5 + 4.33i)T
good7 1+(−14−24.2i)T+(−171.5+297.i)T2 1 + (-14 - 24.2i)T + (-171.5 + 297. i)T^{2}
11 1+(−12−20.7i)T+(−665.5+1.15e3i)T2 1 + (-12 - 20.7i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(−35+60.6i)T+(−1.09e3−1.90e3i)T2 1 + (-35 + 60.6i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1−102T+4.91e3T2 1 - 102T + 4.91e3T^{2}
19 1−20T+6.85e3T2 1 - 20T + 6.85e3T^{2}
23 1+(−36+62.3i)T+(−6.08e3−1.05e4i)T2 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(153+265.i)T+(−1.21e4+2.11e4i)T2 1 + (153 + 265. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(−68+117.i)T+(−1.48e4−2.57e4i)T2 1 + (-68 + 117. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+214T+5.06e4T2 1 + 214T + 5.06e4T^{2}
41 1+(−75+129.i)T+(−3.44e4−5.96e4i)T2 1 + (-75 + 129. i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(−146−252.i)T+(−3.97e4+6.88e4i)T2 1 + (-146 - 252. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−36−62.3i)T+(−5.19e4+8.99e4i)T2 1 + (-36 - 62.3i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+414T+1.48e5T2 1 + 414T + 1.48e5T^{2}
59 1+(−372+644.i)T+(−1.02e5−1.77e5i)T2 1 + (-372 + 644. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−209−361.i)T+(−1.13e5+1.96e5i)T2 1 + (-209 - 361. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(94−162.i)T+(−1.50e5−2.60e5i)T2 1 + (94 - 162. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−480T+3.57e5T2 1 - 480T + 3.57e5T^{2}
73 1−434T+3.89e5T2 1 - 434T + 3.89e5T^{2}
79 1+(676+1.17e3i)T+(−2.46e5+4.26e5i)T2 1 + (676 + 1.17e3i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+(−306−530.i)T+(−2.85e5+4.95e5i)T2 1 + (-306 - 530. i)T + (-2.85e5 + 4.95e5i)T^{2}
89 1+30T+7.04e5T2 1 + 30T + 7.04e5T^{2}
97 1+(−143−247.i)T+(−4.56e5+7.90e5i)T2 1 + (-143 - 247. i)T + (-4.56e5 + 7.90e5i)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

−9.004950288388443492818748560512, −8.038271361946161501970497748082, −7.80420787189111440276171276538, −6.32354982596190584996862678230, −5.59943108236760566824742908738, −5.11547745629316050356351775688, −3.93901412510655582070102294841, −2.81209350492732998515621274386, −1.84197804878582479252471801470, −0.75947991292065423044074001193, 1.05097088206094220398406702252, 1.64826094086859136519938954158, 3.33980812878794745722594568203, 3.84371858473859105265148134215, 4.95052222643614238358329786420, 5.80833882506336601245316338744, 6.91549750502524689844509254828, 7.28341902959714912689409168425, 8.273532226706615999655988689460, 9.058430558841057384794197740581

Graph of the ZZ-function along the critical line