↑

Properties

Label 2-20e2-20.3-c3-0-26
Degree 22
Conductor 400400
Sign −0.997+0.0706i-0.997 + 0.0706i
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  + (6.59 − 6.59i)3-s + (−17.4 − 17.4i)7-s − 59.9i·9-s + 1.16i·11-s + (−32.6 − 32.6i)13-s + (−90.9 + 90.9i)17-s + 131.·19-s − 229.·21-s + (−98.9 + 98.9i)23-s + (−217. − 217. i)27-s − 167. i·29-s + 60.1i·31-s + (7.71 + 7.71i)33-s + (193. − 193. i)37-s − 430.·39-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s + (−0.941 − 0.941i)7-s − 2.22i·9-s + 0.0320i·11-s + (−0.696 − 0.696i)13-s + (−1.29 + 1.29i)17-s + 1.58·19-s − 2.38·21-s + (−0.897 + 0.897i)23-s + (−1.54 − 1.54i)27-s − 1.07i·29-s + 0.348i·31-s + (0.0406 + 0.0406i)33-s + (0.859 − 0.859i)37-s − 1.76·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.997+0.0706i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((−0.997+0.0706i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.997+0.0706i-0.997 + 0.0706i
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), −0.997+0.0706i)(2,\ 400,\ (\ :3/2),\ -0.997 + 0.0706i)

Particular Values

L(2)L(2) ≈\approx 1.8023378001.802337800
L(12)L(\frac12) ≈\approx 1.8023378001.802337800
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+(−6.59+6.59i)T−27iT2 1 + (-6.59 + 6.59i)T - 27iT^{2}
7 1+(17.4+17.4i)T+343iT2 1 + (17.4 + 17.4i)T + 343iT^{2}
11 1−1.16iT−1.33e3T2 1 - 1.16iT - 1.33e3T^{2}
13 1+(32.6+32.6i)T+2.19e3iT2 1 + (32.6 + 32.6i)T + 2.19e3iT^{2}
17 1+(90.9−90.9i)T−4.91e3iT2 1 + (90.9 - 90.9i)T - 4.91e3iT^{2}
19 1−131.T+6.85e3T2 1 - 131.T + 6.85e3T^{2}
23 1+(98.9−98.9i)T−1.21e4iT2 1 + (98.9 - 98.9i)T - 1.21e4iT^{2}
29 1+167.iT−2.43e4T2 1 + 167. iT - 2.43e4T^{2}
31 1−60.1iT−2.97e4T2 1 - 60.1iT - 2.97e4T^{2}
37 1+(−193.+193.i)T−5.06e4iT2 1 + (-193. + 193. i)T - 5.06e4iT^{2}
41 1−28.7T+6.89e4T2 1 - 28.7T + 6.89e4T^{2}
43 1+(−76.6+76.6i)T−7.95e4iT2 1 + (-76.6 + 76.6i)T - 7.95e4iT^{2}
47 1+(176.+176.i)T+1.03e5iT2 1 + (176. + 176. i)T + 1.03e5iT^{2}
53 1+(327.+327.i)T+1.48e5iT2 1 + (327. + 327. i)T + 1.48e5iT^{2}
59 1−141.T+2.05e5T2 1 - 141.T + 2.05e5T^{2}
61 1−369.T+2.26e5T2 1 - 369.T + 2.26e5T^{2}
67 1+(−67.8−67.8i)T+3.00e5iT2 1 + (-67.8 - 67.8i)T + 3.00e5iT^{2}
71 1+46.7iT−3.57e5T2 1 + 46.7iT - 3.57e5T^{2}
73 1+(−141.−141.i)T+3.89e5iT2 1 + (-141. - 141. i)T + 3.89e5iT^{2}
79 1+1.03e3T+4.93e5T2 1 + 1.03e3T + 4.93e5T^{2}
83 1+(−498.+498.i)T−5.71e5iT2 1 + (-498. + 498. i)T - 5.71e5iT^{2}
89 1+1.53e3iT−7.04e5T2 1 + 1.53e3iT - 7.04e5T^{2}
97 1+(1.20−1.20i)T−9.12e5iT2 1 + (1.20 - 1.20i)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

−10.08348746188641124117509816235, −9.480718028812613641308927043989, −8.333910699416726171764187725289, −7.57387053519657637155649609761, −6.93651526024703157531445521275, −5.93707992099467121421854206336, −3.97777359571307356220570967164, −3.09314539331514450679861530567, −1.89470722733848088682460785841, −0.47753897841414488553544882585, 2.44917552105375092220994586503, 3.03144160203693231859556739043, 4.32147215299080993724362621057, 5.19016842886119302402683585690, 6.64657916294049540387242802639, 7.84177895032708379232204941587, 8.962872619063736021308676983288, 9.411348311806986119318865359039, 9.924702737770019727248920041684, 11.17379179429748893888960149783

Graph of the ZZ-function along the critical line