↑

Properties

Label 2-6480-1.1-c1-0-68
Degree 22
Conductor 64806480
Sign −1-1
Analytic cond. 51.743051.7430
Root an. cond. 7.193267.19326
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3.44·7-s + 2·17-s + 6.89·19-s − 7.44·23-s + 25-s + 1.89·29-s − 1.10·31-s − 3.44·35-s − 6·37-s + 9.89·41-s − 11.7·43-s − 9.44·47-s + 4.89·49-s − 7.79·53-s − 1.10·59-s − 3·61-s + 13.2·67-s + 9.79·71-s + 13.7·73-s + 6.89·79-s − 5.44·83-s + 2·85-s − 2.79·89-s + 6.89·95-s + 2·97-s + 2·101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.30·7-s + 0.485·17-s + 1.58·19-s − 1.55·23-s + 0.200·25-s + 0.352·29-s − 0.197·31-s − 0.583·35-s − 0.986·37-s + 1.54·41-s − 1.79·43-s − 1.37·47-s + 0.699·49-s − 1.07·53-s − 0.143·59-s − 0.384·61-s + 1.61·67-s + 1.16·71-s + 1.61·73-s + 0.776·79-s − 0.598·83-s + 0.216·85-s − 0.296·89-s + 0.707·95-s + 0.203·97-s + 0.199·101-s + ⋯

Functional equation

Λ(s)=(6480s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6480s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 64806480    =    24⋅34⋅52^{4} \cdot 3^{4} \cdot 5
Sign: −1-1
Analytic conductor: 51.743051.7430
Root analytic conductor: 7.193267.19326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6480, ( :1/2), −1)(2,\ 6480,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
5 1−T 1 - T
good7 1+3.44T+7T2 1 + 3.44T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−6.89T+19T2 1 - 6.89T + 19T^{2}
23 1+7.44T+23T2 1 + 7.44T + 23T^{2}
29 1−1.89T+29T2 1 - 1.89T + 29T^{2}
31 1+1.10T+31T2 1 + 1.10T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1−9.89T+41T2 1 - 9.89T + 41T^{2}
43 1+11.7T+43T2 1 + 11.7T + 43T^{2}
47 1+9.44T+47T2 1 + 9.44T + 47T^{2}
53 1+7.79T+53T2 1 + 7.79T + 53T^{2}
59 1+1.10T+59T2 1 + 1.10T + 59T^{2}
61 1+3T+61T2 1 + 3T + 61T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1−9.79T+71T2 1 - 9.79T + 71T^{2}
73 1−13.7T+73T2 1 - 13.7T + 73T^{2}
79 1−6.89T+79T2 1 - 6.89T + 79T^{2}
83 1+5.44T+83T2 1 + 5.44T + 83T^{2}
89 1+2.79T+89T2 1 + 2.79T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−7.72625311564088074510738200508, −6.74323440673612004599100064453, −6.37927948760041414273827091542, −5.55024863937292157792583271726, −4.97050102712626707296941390519, −3.72832506110831643065938591819, −3.30580632779638021857339992849, −2.37826302435680562046843035071, −1.29002650284380719422843990297, 0, 1.29002650284380719422843990297, 2.37826302435680562046843035071, 3.30580632779638021857339992849, 3.72832506110831643065938591819, 4.97050102712626707296941390519, 5.55024863937292157792583271726, 6.37927948760041414273827091542, 6.74323440673612004599100064453, 7.72625311564088074510738200508

Graph of the ZZ-function along the critical line