↑

Properties

Label 2-3800-5.4-c1-0-76
Degree 22
Conductor 38003800
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
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  − 2.32i·3-s − 1.39i·7-s − 2.39·9-s − 4.32i·13-s + 0.601i·17-s − 19-s − 3.24·21-s − 6.04i·23-s − 1.39i·27-s − 4.60·29-s + 2.79·31-s + 1.07i·37-s − 10.0·39-s − 5.44·41-s + 8.64i·43-s + ⋯
L(s)  = 1  − 1.34i·3-s − 0.528i·7-s − 0.799·9-s − 1.19i·13-s + 0.145i·17-s − 0.229·19-s − 0.708·21-s − 1.26i·23-s − 0.269i·27-s − 0.854·29-s + 0.502·31-s + 0.176i·37-s − 1.60·39-s − 0.850·41-s + 1.31i·43-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3800(3649,⋅)\chi_{3800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), −0.894−0.447i)(2,\ 3800,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.1609781111.160978111
L(12)L(\frac12) ≈\approx 1.1609781111.160978111
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
5 1 1
19 1+T 1 + T
good3 1+2.32iT−3T2 1 + 2.32iT - 3T^{2}
7 1+1.39iT−7T2 1 + 1.39iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4.32iT−13T2 1 + 4.32iT - 13T^{2}
17 1−0.601iT−17T2 1 - 0.601iT - 17T^{2}
23 1+6.04iT−23T2 1 + 6.04iT - 23T^{2}
29 1+4.60T+29T2 1 + 4.60T + 29T^{2}
31 1−2.79T+31T2 1 - 2.79T + 31T^{2}
37 1−1.07iT−37T2 1 - 1.07iT - 37T^{2}
41 1+5.44T+41T2 1 + 5.44T + 41T^{2}
43 1−8.64iT−43T2 1 - 8.64iT - 43T^{2}
47 1+1.85iT−47T2 1 + 1.85iT - 47T^{2}
53 1−3.11iT−53T2 1 - 3.11iT - 53T^{2}
59 1−6.69T+59T2 1 - 6.69T + 59T^{2}
61 1+2.64T+61T2 1 + 2.64T + 61T^{2}
67 1+14.4iT−67T2 1 + 14.4iT - 67T^{2}
71 1+5.59T+71T2 1 + 5.59T + 71T^{2}
73 1+12.6iT−73T2 1 + 12.6iT - 73T^{2}
79 1−4.64T+79T2 1 - 4.64T + 79T^{2}
83 1−1.20iT−83T2 1 - 1.20iT - 83T^{2}
89 1+9.44T+89T2 1 + 9.44T + 89T^{2}
97 1−4.51iT−97T2 1 - 4.51iT - 97T^{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

−7.963064896663921291227471183231, −7.40843104755245891182732277941, −6.61540710152001833228520760373, −6.12045306858802990808457147828, −5.17386682172516514936360606494, −4.23071940256840709222236083897, −3.16697465462797393106678710151, −2.30617675650838253723635086167, −1.27309282765583189636570105611, −0.34400869778399379051097310846, 1.64587361621402961688956936471, 2.71488540180933286473424904123, 3.79803893725793442786446647294, 4.18499771242696888959210147481, 5.21700491204007778372766033472, 5.62427070833614467710536656700, 6.72572131064042491774303449444, 7.39728571894688107367864330328, 8.541826102884353559587103949550, 8.979844754746252746078036013071

Graph of the ZZ-function along the critical line