↑

Properties

Label 2-336-12.11-c5-0-19
Degree 22
Conductor 336336
Sign −0.877−0.478i-0.877 - 0.478i
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
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  + (−7.46 + 13.6i)3-s + 31.0i·5-s + 49i·7-s + (−131. − 204. i)9-s + 273.·11-s + 431.·13-s + (−424. − 231. i)15-s + 648. i·17-s + 2.30e3i·19-s + (−670. − 365. i)21-s + 3.72e3·23-s + 2.16e3·25-s + (3.77e3 − 274. i)27-s + 7.00e3i·29-s − 1.72e3i·31-s + ⋯
L(s)  = 1  + (−0.478 + 0.877i)3-s + 0.554i·5-s + 0.377i·7-s + (−0.541 − 0.840i)9-s + 0.680·11-s + 0.708·13-s + (−0.486 − 0.265i)15-s + 0.544i·17-s + 1.46i·19-s + (−0.331 − 0.181i)21-s + 1.46·23-s + 0.692·25-s + (0.997 − 0.0723i)27-s + 1.54i·29-s − 0.322i·31-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((−0.877−0.478i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=((−0.877−0.478i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −0.877−0.478i-0.877 - 0.478i
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ336(239,⋅)\chi_{336} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 336, ( :5/2), −0.877−0.478i)(2,\ 336,\ (\ :5/2),\ -0.877 - 0.478i)

Particular Values

L(3)L(3) ≈\approx 1.5998929951.599892995
L(12)L(\frac12) ≈\approx 1.5998929951.599892995
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 1
3 1+(7.46−13.6i)T 1 + (7.46 - 13.6i)T
7 1−49iT 1 - 49iT
good5 1−31.0iT−3.12e3T2 1 - 31.0iT - 3.12e3T^{2}
11 1−273.T+1.61e5T2 1 - 273.T + 1.61e5T^{2}
13 1−431.T+3.71e5T2 1 - 431.T + 3.71e5T^{2}
17 1−648.iT−1.41e6T2 1 - 648. iT - 1.41e6T^{2}
19 1−2.30e3iT−2.47e6T2 1 - 2.30e3iT - 2.47e6T^{2}
23 1−3.72e3T+6.43e6T2 1 - 3.72e3T + 6.43e6T^{2}
29 1−7.00e3iT−2.05e7T2 1 - 7.00e3iT - 2.05e7T^{2}
31 1+1.72e3iT−2.86e7T2 1 + 1.72e3iT - 2.86e7T^{2}
37 1+1.22e4T+6.93e7T2 1 + 1.22e4T + 6.93e7T^{2}
41 1+1.66e4iT−1.15e8T2 1 + 1.66e4iT - 1.15e8T^{2}
43 1−9.18e3iT−1.47e8T2 1 - 9.18e3iT - 1.47e8T^{2}
47 1−2.32e4T+2.29e8T2 1 - 2.32e4T + 2.29e8T^{2}
53 1−1.91e4iT−4.18e8T2 1 - 1.91e4iT - 4.18e8T^{2}
59 1−1.03e4T+7.14e8T2 1 - 1.03e4T + 7.14e8T^{2}
61 1+7.05e3T+8.44e8T2 1 + 7.05e3T + 8.44e8T^{2}
67 1+4.61e4iT−1.35e9T2 1 + 4.61e4iT - 1.35e9T^{2}
71 1+7.59e4T+1.80e9T2 1 + 7.59e4T + 1.80e9T^{2}
73 1+5.11e4T+2.07e9T2 1 + 5.11e4T + 2.07e9T^{2}
79 1−1.06e4iT−3.07e9T2 1 - 1.06e4iT - 3.07e9T^{2}
83 1−6.75e4T+3.93e9T2 1 - 6.75e4T + 3.93e9T^{2}
89 1−7.33e4iT−5.58e9T2 1 - 7.33e4iT - 5.58e9T^{2}
97 1+1.34e5T+8.58e9T2 1 + 1.34e5T + 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

−10.73486720385611442455124385419, −10.58210909759988946993360758069, −9.190989470223700092415579688826, −8.649397589502309823894958726179, −7.09357917006337208741881973573, −6.14564600161974753559646952875, −5.27846358951135628295567176978, −3.95739494579378857883922664173, −3.13325638942362249827475710812, −1.31613767352388530754515189236, 0.51336832573608185852432090710, 1.31483080086695799086820037546, 2.84208721762975740231967984598, 4.45371700839426727125233820058, 5.41524465824240818456266116270, 6.63835599994528283945214646074, 7.22327280798519055389670522057, 8.502476597024468503012514956317, 9.166075354273378441828158262232, 10.57934306398198466200771466755

Graph of the ZZ-function along the critical line