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Properties

Label 2-2268-1.1-c1-0-7
Degree 22
Conductor 22682268
Sign 11
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  + 1.11·5-s + 7-s + 1.88·11-s − 13-s + 5.87·17-s + 7.09·19-s − 3.98·23-s − 3.76·25-s − 0.987·29-s − 0.666·31-s + 1.11·35-s − 1.33·37-s + 1.88·41-s − 10.8·43-s + 11.0·47-s + 49-s + 12.2·53-s + 2.09·55-s − 4.76·59-s + 3.76·61-s − 1.11·65-s + 4.09·67-s + 10.7·71-s − 3.09·73-s + 1.88·77-s + 6.43·79-s + 11.8·83-s + ⋯
L(s)  = 1  + 0.496·5-s + 0.377·7-s + 0.569·11-s − 0.277·13-s + 1.42·17-s + 1.62·19-s − 0.831·23-s − 0.753·25-s − 0.183·29-s − 0.119·31-s + 0.187·35-s − 0.219·37-s + 0.294·41-s − 1.65·43-s + 1.61·47-s + 0.142·49-s + 1.67·53-s + 0.283·55-s − 0.620·59-s + 0.482·61-s − 0.137·65-s + 0.500·67-s + 1.27·71-s − 0.362·73-s + 0.215·77-s + 0.723·79-s + 1.30·83-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 11
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 1)(2,\ 2268,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2534137942.253413794
L(12)L(\frac12) ≈\approx 2.2534137942.253413794
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
7 1−T 1 - T
good5 1−1.11T+5T2 1 - 1.11T + 5T^{2}
11 1−1.88T+11T2 1 - 1.88T + 11T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1−5.87T+17T2 1 - 5.87T + 17T^{2}
19 1−7.09T+19T2 1 - 7.09T + 19T^{2}
23 1+3.98T+23T2 1 + 3.98T + 23T^{2}
29 1+0.987T+29T2 1 + 0.987T + 29T^{2}
31 1+0.666T+31T2 1 + 0.666T + 31T^{2}
37 1+1.33T+37T2 1 + 1.33T + 37T^{2}
41 1−1.88T+41T2 1 - 1.88T + 41T^{2}
43 1+10.8T+43T2 1 + 10.8T + 43T^{2}
47 1−11.0T+47T2 1 - 11.0T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1+4.76T+59T2 1 + 4.76T + 59T^{2}
61 1−3.76T+61T2 1 - 3.76T + 61T^{2}
67 1−4.09T+67T2 1 - 4.09T + 67T^{2}
71 1−10.7T+71T2 1 - 10.7T + 71T^{2}
73 1+3.09T+73T2 1 + 3.09T + 73T^{2}
79 1−6.43T+79T2 1 - 6.43T + 79T^{2}
83 1−11.8T+83T2 1 - 11.8T + 83T^{2}
89 1+14.3T+89T2 1 + 14.3T + 89T^{2}
97 1−0.765T+97T2 1 - 0.765T + 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

−9.170253474368430120861992693460, −8.137358786635108106077005880484, −7.55990863272468300779993836038, −6.73661088880707328437725493218, −5.62765808299818400022755349188, −5.35072332053585461880387847954, −4.08227221565028679776404096391, −3.27460322272164033596761099256, −2.07474786433568748434270171766, −1.04483332189833138687378324111, 1.04483332189833138687378324111, 2.07474786433568748434270171766, 3.27460322272164033596761099256, 4.08227221565028679776404096391, 5.35072332053585461880387847954, 5.62765808299818400022755349188, 6.73661088880707328437725493218, 7.55990863272468300779993836038, 8.137358786635108106077005880484, 9.170253474368430120861992693460

Graph of the ZZ-function along the critical line