↑

Properties

Label 2-6900-5.4-c1-0-19
Degree 22
Conductor 69006900
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 55.096755.0967
Root an. cond. 7.422727.42272
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  + i·3-s + i·7-s − 9-s − 4.44·11-s − 2.44i·13-s − 3.44i·17-s − 7.34·19-s − 21-s − i·23-s − i·27-s + 9.44·29-s − 1.89·31-s − 4.44i·33-s − 9.89i·37-s + 2.44·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.34·11-s − 0.679i·13-s − 0.836i·17-s − 1.68·19-s − 0.218·21-s − 0.208i·23-s − 0.192i·27-s + 1.75·29-s − 0.341·31-s − 0.774i·33-s − 1.62i·37-s + 0.392·39-s + ⋯

Functional equation

Λ(s)=(6900s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6900s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 69006900    =    22⋅3⋅52⋅232^{2} \cdot 3 \cdot 5^{2} \cdot 23
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 55.096755.0967
Root analytic conductor: 7.422727.42272
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6900(6349,⋅)\chi_{6900} (6349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6900, ( :1/2), 0.447−0.894i)(2,\ 6900,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.2799567111.279956711
L(12)L(\frac12) ≈\approx 1.2799567111.279956711
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
3 1−iT 1 - iT
5 1 1
23 1+iT 1 + iT
good7 1−iT−7T2 1 - iT - 7T^{2}
11 1+4.44T+11T2 1 + 4.44T + 11T^{2}
13 1+2.44iT−13T2 1 + 2.44iT - 13T^{2}
17 1+3.44iT−17T2 1 + 3.44iT - 17T^{2}
19 1+7.34T+19T2 1 + 7.34T + 19T^{2}
29 1−9.44T+29T2 1 - 9.44T + 29T^{2}
31 1+1.89T+31T2 1 + 1.89T + 31T^{2}
37 1+9.89iT−37T2 1 + 9.89iT - 37T^{2}
41 1+0.550T+41T2 1 + 0.550T + 41T^{2}
43 1−7.79iT−43T2 1 - 7.79iT - 43T^{2}
47 1−7.34iT−47T2 1 - 7.34iT - 47T^{2}
53 1+4.34iT−53T2 1 + 4.34iT - 53T^{2}
59 1−6.55T+59T2 1 - 6.55T + 59T^{2}
61 1−0.449T+61T2 1 - 0.449T + 61T^{2}
67 1−8.79iT−67T2 1 - 8.79iT - 67T^{2}
71 1−2.34T+71T2 1 - 2.34T + 71T^{2}
73 1−7.34iT−73T2 1 - 7.34iT - 73T^{2}
79 1−13.7T+79T2 1 - 13.7T + 79T^{2}
83 1−15.4iT−83T2 1 - 15.4iT - 83T^{2}
89 1−3.10T+89T2 1 - 3.10T + 89T^{2}
97 1−4.89iT−97T2 1 - 4.89iT - 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

−8.250218229261969247986287211029, −7.48798000741704326785257263251, −6.61695438552252573796770840184, −5.85163458118044833736888380961, −5.19185113491397915728215025493, −4.61442598403851442222482269322, −3.76817378375079348899863129121, −2.64597741395011625175523822498, −2.42503599151958582720818296757, −0.68502340513980062447050138675, 0.45263153025545721513988502309, 1.75624935628303556591469966681, 2.40632371359559877412702427040, 3.36994151581093976930997991288, 4.31191390854305006432098092524, 4.94991970835248592635296359697, 5.85823240860611581496770586435, 6.55181880838343658152915948167, 7.03794079556070990732760205121, 7.947069103211280545808572967565

Graph of the ZZ-function along the critical line