↑

Properties

Label 2-1216-152.107-c1-0-7
Degree 22
Conductor 12161216
Sign −0.304−0.952i-0.304 - 0.952i
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
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  + (0.121 − 0.0702i)3-s + (1.5 − 0.866i)5-s + 2i·7-s + (−1.49 + 2.58i)9-s + (−2.13 + 3.69i)13-s + (0.121 − 0.210i)15-s + (1.99 + 3.44i)17-s + (−4.31 + 0.630i)19-s + (0.140 + 0.243i)21-s + (−6.40 − 3.70i)23-s + (−1 + 1.73i)25-s + 0.839i·27-s + (1.99 − 3.44i)29-s − 8.62·31-s + (1.73 + 3i)35-s + ⋯
L(s)  = 1  + (0.0702 − 0.0405i)3-s + (0.670 − 0.387i)5-s + 0.755i·7-s + (−0.496 + 0.860i)9-s + (−0.590 + 1.02i)13-s + (0.0314 − 0.0543i)15-s + (0.482 + 0.836i)17-s + (−0.989 + 0.144i)19-s + (0.0306 + 0.0530i)21-s + (−1.33 − 0.771i)23-s + (−0.200 + 0.346i)25-s + 0.161i·27-s + (0.369 − 0.640i)29-s − 1.54·31-s + (0.292 + 0.507i)35-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=((−0.304−0.952i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=((−0.304−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: −0.304−0.952i-0.304 - 0.952i
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1216(31,⋅)\chi_{1216} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1216, ( :1/2), −0.304−0.952i)(2,\ 1216,\ (\ :1/2),\ -0.304 - 0.952i)

Particular Values

L(1)L(1) ≈\approx 1.2356530741.235653074
L(12)L(\frac12) ≈\approx 1.2356530741.235653074
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
19 1+(4.31−0.630i)T 1 + (4.31 - 0.630i)T
good3 1+(−0.121+0.0702i)T+(1.5−2.59i)T2 1 + (-0.121 + 0.0702i)T + (1.5 - 2.59i)T^{2}
5 1+(−1.5+0.866i)T+(2.5−4.33i)T2 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+(2.13−3.69i)T+(−6.5−11.2i)T2 1 + (2.13 - 3.69i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1.99−3.44i)T+(−8.5+14.7i)T2 1 + (-1.99 - 3.44i)T + (-8.5 + 14.7i)T^{2}
23 1+(6.40+3.70i)T+(11.5+19.9i)T2 1 + (6.40 + 3.70i)T + (11.5 + 19.9i)T^{2}
29 1+(−1.99+3.44i)T+(−14.5−25.1i)T2 1 + (-1.99 + 3.44i)T + (-14.5 - 25.1i)T^{2}
31 1+8.62T+31T2 1 + 8.62T + 31T^{2}
37 1−7.26T+37T2 1 - 7.26T + 37T^{2}
41 1+(6.39−3.69i)T+(20.5−35.5i)T2 1 + (6.39 - 3.69i)T + (20.5 - 35.5i)T^{2}
43 1+(−6.16−10.6i)T+(−21.5+37.2i)T2 1 + (-6.16 - 10.6i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.364−0.210i)T+(23.5+40.7i)T2 1 + (-0.364 - 0.210i)T + (23.5 + 40.7i)T^{2}
53 1+(−5.48+9.49i)T+(−26.5−45.8i)T2 1 + (-5.48 + 9.49i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.21−0.700i)T+(29.5−51.0i)T2 1 + (1.21 - 0.700i)T + (29.5 - 51.0i)T^{2}
61 1+(1.92+1.10i)T+(30.5+52.8i)T2 1 + (1.92 + 1.10i)T + (30.5 + 52.8i)T^{2}
67 1+(1.81+1.05i)T+(33.5+58.0i)T2 1 + (1.81 + 1.05i)T + (33.5 + 58.0i)T^{2}
71 1+(0.970+1.68i)T+(−35.5+61.4i)T2 1 + (0.970 + 1.68i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.5−4.33i)T+(−36.5+63.2i)T2 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2.70−4.68i)T+(−39.5+68.4i)T2 1 + (-2.70 - 4.68i)T + (-39.5 + 68.4i)T^{2}
83 1−8.62T+83T2 1 - 8.62T + 83T^{2}
89 1+(−14.5−8.39i)T+(44.5+77.0i)T2 1 + (-14.5 - 8.39i)T + (44.5 + 77.0i)T^{2}
97 1+(6.39−3.69i)T+(48.5−84.0i)T2 1 + (6.39 - 3.69i)T + (48.5 - 84.0i)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

−9.851964274219244456106558781603, −9.180562398015760317858407738593, −8.363452894721871408843627462555, −7.74493215551036837809081104458, −6.40304955514082621228131980893, −5.83821929344037986243276536383, −4.95304807214290697016606340705, −3.99321090581671780475550468438, −2.41000041102627501412029117061, −1.87451264252166576248107898231, 0.48234309365699207345639393880, 2.16951257146972139665536456607, 3.23314402344832794208202516936, 4.14138259476655740867111786322, 5.45822027909683206502999475365, 6.04005234180005582847275140623, 7.08295975818626780033169200325, 7.70750281246000047428329364421, 8.800978555214558385541652872765, 9.574045001007696320701798444410

Graph of the ZZ-function along the critical line