↑

Properties

Label 2-85e2-1.1-c1-0-111
Degree 22
Conductor 72257225
Sign 11
Analytic cond. 57.691957.6919
Root an. cond. 7.595517.59551
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·2-s + 0.334·3-s − 0.302·4-s + 0.436·6-s + 3.90·7-s − 2.99·8-s − 2.88·9-s − 0.135·11-s − 0.101·12-s − 6.65·13-s + 5.08·14-s − 3.30·16-s − 3.76·18-s − 5.16·19-s + 1.30·21-s − 0.176·22-s + 4.32·23-s − 1.00·24-s − 8.67·26-s − 1.97·27-s − 1.18·28-s − 2.14·29-s + 8.61·31-s + 1.69·32-s − 0.0453·33-s + 0.873·36-s + 0.234·37-s + ⋯
L(s)  = 1  + 0.921·2-s + 0.193·3-s − 0.151·4-s + 0.178·6-s + 1.47·7-s − 1.06·8-s − 0.962·9-s − 0.0408·11-s − 0.0292·12-s − 1.84·13-s + 1.35·14-s − 0.825·16-s − 0.886·18-s − 1.18·19-s + 0.285·21-s − 0.0376·22-s + 0.901·23-s − 0.204·24-s − 1.70·26-s − 0.379·27-s − 0.223·28-s − 0.398·29-s + 1.54·31-s + 0.299·32-s − 0.00789·33-s + 0.145·36-s + 0.0385·37-s + ⋯

Functional equation

Λ(s)=(7225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72257225    =    52⋅1725^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 57.691957.6919
Root analytic conductor: 7.595517.59551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7225, ( :1/2), 1)(2,\ 7225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5713786932.571378693
L(12)L(\frac12) ≈\approx 2.5713786932.571378693
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)
bad5 1 1
17 1 1
good2 1−1.30T+2T2 1 - 1.30T + 2T^{2}
3 1−0.334T+3T2 1 - 0.334T + 3T^{2}
7 1−3.90T+7T2 1 - 3.90T + 7T^{2}
11 1+0.135T+11T2 1 + 0.135T + 11T^{2}
13 1+6.65T+13T2 1 + 6.65T + 13T^{2}
19 1+5.16T+19T2 1 + 5.16T + 19T^{2}
23 1−4.32T+23T2 1 - 4.32T + 23T^{2}
29 1+2.14T+29T2 1 + 2.14T + 29T^{2}
31 1−8.61T+31T2 1 - 8.61T + 31T^{2}
37 1−0.234T+37T2 1 - 0.234T + 37T^{2}
41 1−8.34T+41T2 1 - 8.34T + 41T^{2}
43 1−1.89T+43T2 1 - 1.89T + 43T^{2}
47 1−9.86T+47T2 1 - 9.86T + 47T^{2}
53 1−1.42T+53T2 1 - 1.42T + 53T^{2}
59 1+3.61T+59T2 1 + 3.61T + 59T^{2}
61 1−5.46T+61T2 1 - 5.46T + 61T^{2}
67 1−8.10T+67T2 1 - 8.10T + 67T^{2}
71 1−10.4T+71T2 1 - 10.4T + 71T^{2}
73 1−1.26T+73T2 1 - 1.26T + 73T^{2}
79 1−12.1T+79T2 1 - 12.1T + 79T^{2}
83 1−1.43T+83T2 1 - 1.43T + 83T^{2}
89 1+4.30T+89T2 1 + 4.30T + 89T^{2}
97 1+0.473T+97T2 1 + 0.473T + 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

−7.966728621024898697513123689607, −7.25875699376531620879299883270, −6.32363734023026436246715301019, −5.55693312709260858433327089189, −4.91031319722251061385475076273, −4.59421317548806032145256235744, −3.73981605282311589289469828327, −2.53247656946161313461513806648, −2.35417698448044852656091346311, −0.67362848496811495240874602182, 0.67362848496811495240874602182, 2.35417698448044852656091346311, 2.53247656946161313461513806648, 3.73981605282311589289469828327, 4.59421317548806032145256235744, 4.91031319722251061385475076273, 5.55693312709260858433327089189, 6.32363734023026436246715301019, 7.25875699376531620879299883270, 7.966728621024898697513123689607

Graph of the ZZ-function along the critical line