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Properties

Label 2-8280-1.1-c1-0-55
Degree 22
Conductor 82808280
Sign 11
Analytic cond. 66.116166.1161
Root an. cond. 8.131188.13118
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4.73·7-s + 0.360·11-s + 5.26·13-s − 0.370·17-s − 4.60·19-s + 23-s + 25-s − 0.939·29-s + 9.66·31-s + 4.73·35-s + 3.26·37-s − 5.29·41-s + 1.25·47-s + 15.4·49-s − 10.9·53-s + 0.360·55-s + 9.66·59-s + 9.71·61-s + 5.26·65-s − 7.07·67-s − 11.3·71-s + 0.745·73-s + 1.70·77-s + 0.415·79-s + 9.26·83-s − 0.370·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.79·7-s + 0.108·11-s + 1.45·13-s − 0.0899·17-s − 1.05·19-s + 0.208·23-s + 0.200·25-s − 0.174·29-s + 1.73·31-s + 0.800·35-s + 0.537·37-s − 0.827·41-s + 0.183·47-s + 2.20·49-s − 1.50·53-s + 0.0486·55-s + 1.25·59-s + 1.24·61-s + 0.652·65-s − 0.863·67-s − 1.34·71-s + 0.0872·73-s + 0.194·77-s + 0.0467·79-s + 1.01·83-s − 0.0402·85-s + ⋯

Functional equation

Λ(s)=(8280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82808280    =    23⋅32⋅5⋅232^{3} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 66.116166.1161
Root analytic conductor: 8.131188.13118
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8280, ( :1/2), 1)(2,\ 8280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3430736003.343073600
L(12)L(\frac12) ≈\approx 3.3430736003.343073600
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
23 1−T 1 - T
good7 1−4.73T+7T2 1 - 4.73T + 7T^{2}
11 1−0.360T+11T2 1 - 0.360T + 11T^{2}
13 1−5.26T+13T2 1 - 5.26T + 13T^{2}
17 1+0.370T+17T2 1 + 0.370T + 17T^{2}
19 1+4.60T+19T2 1 + 4.60T + 19T^{2}
29 1+0.939T+29T2 1 + 0.939T + 29T^{2}
31 1−9.66T+31T2 1 - 9.66T + 31T^{2}
37 1−3.26T+37T2 1 - 3.26T + 37T^{2}
41 1+5.29T+41T2 1 + 5.29T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1−1.25T+47T2 1 - 1.25T + 47T^{2}
53 1+10.9T+53T2 1 + 10.9T + 53T^{2}
59 1−9.66T+59T2 1 - 9.66T + 59T^{2}
61 1−9.71T+61T2 1 - 9.71T + 61T^{2}
67 1+7.07T+67T2 1 + 7.07T + 67T^{2}
71 1+11.3T+71T2 1 + 11.3T + 71T^{2}
73 1−0.745T+73T2 1 - 0.745T + 73T^{2}
79 1−0.415T+79T2 1 - 0.415T + 79T^{2}
83 1−9.26T+83T2 1 - 9.26T + 83T^{2}
89 1+12.6T+89T2 1 + 12.6T + 89T^{2}
97 1−14.0T+97T2 1 - 14.0T + 97T^{2}
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   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

−8.025934477765864123626538632131, −7.12701054666033135049347327987, −6.30345242303854103628117809745, −5.81566870805209977326073590116, −4.87222448308005404242540730084, −4.45637650079522595400150967894, −3.58958670578191695976521264071, −2.46755196347086650262562836273, −1.68837990145725578811305610802, −0.990846893053214022973120212946, 0.990846893053214022973120212946, 1.68837990145725578811305610802, 2.46755196347086650262562836273, 3.58958670578191695976521264071, 4.45637650079522595400150967894, 4.87222448308005404242540730084, 5.81566870805209977326073590116, 6.30345242303854103628117809745, 7.12701054666033135049347327987, 8.025934477765864123626538632131

Graph of the ZZ-function along the critical line