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Properties

Label 2-2268-1.1-c1-0-1
Degree 22
Conductor 22682268
Sign 11
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 11-s + 2·17-s + 2·19-s − 4·23-s − 25-s − 2·29-s + 6·31-s + 2·35-s + 5·37-s − 6·41-s + 5·43-s + 49-s + 9·53-s + 2·55-s − 2·59-s + 7·67-s + 11·71-s + 4·73-s + 77-s + 15·79-s − 10·83-s − 4·85-s + 12·89-s − 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 0.301·11-s + 0.485·17-s + 0.458·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.07·31-s + 0.338·35-s + 0.821·37-s − 0.937·41-s + 0.762·43-s + 1/7·49-s + 1.23·53-s + 0.269·55-s − 0.260·59-s + 0.855·67-s + 1.30·71-s + 0.468·73-s + 0.113·77-s + 1.68·79-s − 1.09·83-s − 0.433·85-s + 1.27·89-s − 0.410·95-s + 1.01·97-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: yes
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 1.2121962111.212196211
L(12)L(\frac12) ≈\approx 1.2121962111.212196211
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+2T+pT2 1 + 2 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+2T+pT2 1 + 2 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1−11T+pT2 1 - 11 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1+10T+pT2 1 + 10 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−8.997685205994559965713133471010, −8.018426868325578676881599365055, −7.70653582974649011268972075502, −6.73797830713651472136193017201, −5.91237661522044270287152415596, −5.01052820129027652717089088341, −4.03422828904062778172858859761, −3.36799044751201127158795412066, −2.27046415066059517749502915832, −0.70080249099287424881513705202, 0.70080249099287424881513705202, 2.27046415066059517749502915832, 3.36799044751201127158795412066, 4.03422828904062778172858859761, 5.01052820129027652717089088341, 5.91237661522044270287152415596, 6.73797830713651472136193017201, 7.70653582974649011268972075502, 8.018426868325578676881599365055, 8.997685205994559965713133471010

Graph of the ZZ-function along the critical line