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Properties

Label 2-6336-1.1-c1-0-30
Degree 22
Conductor 63366336
Sign 11
Analytic cond. 50.593250.5932
Root an. cond. 7.112897.11289
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  − 3·5-s + 4·7-s − 11-s + 2·13-s + 8·17-s + 6·19-s + 5·23-s + 4·25-s + 4·29-s − 31-s − 12·35-s − 3·37-s + 6·41-s − 6·43-s − 12·47-s + 9·49-s − 6·53-s + 3·55-s − 3·59-s − 6·65-s + 11·67-s − 5·71-s − 10·73-s − 4·77-s + 2·79-s + 2·83-s − 24·85-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s − 0.301·11-s + 0.554·13-s + 1.94·17-s + 1.37·19-s + 1.04·23-s + 4/5·25-s + 0.742·29-s − 0.179·31-s − 2.02·35-s − 0.493·37-s + 0.937·41-s − 0.914·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.404·55-s − 0.390·59-s − 0.744·65-s + 1.34·67-s − 0.593·71-s − 1.17·73-s − 0.455·77-s + 0.225·79-s + 0.219·83-s − 2.60·85-s + ⋯

Functional equation

Λ(s)=(6336s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6336s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63366336    =    26⋅32⋅112^{6} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 50.593250.5932
Root analytic conductor: 7.112897.11289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6336, ( :1/2), 1)(2,\ 6336,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1965712492.196571249
L(12)L(\frac12) ≈\approx 2.1965712492.196571249
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
11 1+T 1 + T
good5 1+3T+pT2 1 + 3 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−5T+pT2 1 - 5 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1−5T+pT2 1 - 5 T + p T^{2}
97 1−13T+pT2 1 - 13 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

−7.956934014776591607573900251598, −7.61429997405739391186912692284, −6.87418988730690416783672401623, −5.68424301869154296654936031860, −5.06592095202887449192765721037, −4.53711525064349731406861939231, −3.48294779280892225932738048713, −3.09318696203506207589934529729, −1.56276573574415865649083879116, −0.856353036609411667859710468731, 0.856353036609411667859710468731, 1.56276573574415865649083879116, 3.09318696203506207589934529729, 3.48294779280892225932738048713, 4.53711525064349731406861939231, 5.06592095202887449192765721037, 5.68424301869154296654936031860, 6.87418988730690416783672401623, 7.61429997405739391186912692284, 7.956934014776591607573900251598

Graph of the ZZ-function along the critical line