↑

Properties

Label 2-20e2-20.3-c3-0-11
Degree 22
Conductor 400400
Sign 0.999−0.0299i0.999 - 0.0299i
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
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  + (−4.12 + 4.12i)3-s + (−4.12 − 4.12i)7-s − 7i·9-s + 13.8i·11-s + (−57.1 − 57.1i)13-s + (57.1 − 57.1i)17-s + 96.9·19-s + 34·21-s + (−86.5 + 86.5i)23-s + (−82.4 − 82.4i)27-s − 174i·29-s + 193. i·31-s + (−57.1 − 57.1i)33-s + 471.·39-s + 252·41-s + ⋯
L(s)  = 1  + (−0.793 + 0.793i)3-s + (−0.222 − 0.222i)7-s − 0.259i·9-s + 0.379i·11-s + (−1.21 − 1.21i)13-s + (0.815 − 0.815i)17-s + 1.17·19-s + 0.353·21-s + (−0.784 + 0.784i)23-s + (−0.587 − 0.587i)27-s − 1.11i·29-s + 1.12i·31-s + (−0.301 − 0.301i)33-s + 1.93·39-s + 0.959·41-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.999−0.0299i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((0.999−0.0299i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.999−0.0299i0.999 - 0.0299i
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ400(143,⋅)\chi_{400} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), 0.999−0.0299i)(2,\ 400,\ (\ :3/2),\ 0.999 - 0.0299i)

Particular Values

L(2)L(2) ≈\approx 1.1376702891.137670289
L(12)L(\frac12) ≈\approx 1.1376702891.137670289
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
5 1 1
good3 1+(4.12−4.12i)T−27iT2 1 + (4.12 - 4.12i)T - 27iT^{2}
7 1+(4.12+4.12i)T+343iT2 1 + (4.12 + 4.12i)T + 343iT^{2}
11 1−13.8iT−1.33e3T2 1 - 13.8iT - 1.33e3T^{2}
13 1+(57.1+57.1i)T+2.19e3iT2 1 + (57.1 + 57.1i)T + 2.19e3iT^{2}
17 1+(−57.1+57.1i)T−4.91e3iT2 1 + (-57.1 + 57.1i)T - 4.91e3iT^{2}
19 1−96.9T+6.85e3T2 1 - 96.9T + 6.85e3T^{2}
23 1+(86.5−86.5i)T−1.21e4iT2 1 + (86.5 - 86.5i)T - 1.21e4iT^{2}
29 1+174iT−2.43e4T2 1 + 174iT - 2.43e4T^{2}
31 1−193.iT−2.97e4T2 1 - 193. iT - 2.97e4T^{2}
37 1−5.06e4iT2 1 - 5.06e4iT^{2}
41 1−252T+6.89e4T2 1 - 252T + 6.89e4T^{2}
43 1+(−202.+202.i)T−7.95e4iT2 1 + (-202. + 202. i)T - 7.95e4iT^{2}
47 1+(−284.−284.i)T+1.03e5iT2 1 + (-284. - 284. i)T + 1.03e5iT^{2}
53 1+(−399.−399.i)T+1.48e5iT2 1 + (-399. - 399. i)T + 1.48e5iT^{2}
59 1−872.T+2.05e5T2 1 - 872.T + 2.05e5T^{2}
61 1−56T+2.26e5T2 1 - 56T + 2.26e5T^{2}
67 1+(317.+317.i)T+3.00e5iT2 1 + (317. + 317. i)T + 3.00e5iT^{2}
71 1−387.iT−3.57e5T2 1 - 387. iT - 3.57e5T^{2}
73 1+(−399.−399.i)T+3.89e5iT2 1 + (-399. - 399. i)T + 3.89e5iT^{2}
79 1−692.T+4.93e5T2 1 - 692.T + 4.93e5T^{2}
83 1+(−482.+482.i)T−5.71e5iT2 1 + (-482. + 482. i)T - 5.71e5iT^{2}
89 1−42iT−7.04e5T2 1 - 42iT - 7.04e5T^{2}
97 1+(−742.+742.i)T−9.12e5iT2 1 + (-742. + 742. i)T - 9.12e5iT^{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

−10.68216213906149723822769245919, −9.967957558908453862088550032969, −9.511933513366028345023304640152, −7.85338255611337026765345260785, −7.23779047412895047788286106579, −5.62835506009130131668222763975, −5.22500044995922650524957014862, −4.01479191856703598805837368881, −2.68788006440449560677194163062, −0.60115872762248519383357366364, 0.861274524639330186916238249525, 2.27932636958682651706552434294, 3.87076171167397399572503674270, 5.28077933714873573659792501898, 6.12278640708764119490572174552, 7.00623038333332482084730903801, 7.81913982425510558950411916909, 9.118063102231774951482887046528, 9.925959266229248336717263118807, 11.06292863118396382221597633199

Graph of the ZZ-function along the critical line