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Properties

Label 2-6336-1.1-c1-0-41
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  + 5-s + 4·7-s + 11-s + 2·13-s + 2·19-s − 9·23-s − 4·25-s + 4·29-s + 5·31-s + 4·35-s + 9·37-s − 2·41-s + 6·43-s + 4·47-s + 9·49-s − 6·53-s + 55-s − 5·59-s + 2·65-s + 13·67-s + 71-s + 14·73-s + 4·77-s − 10·79-s + 14·83-s + 13·89-s + 8·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 1.87·23-s − 4/5·25-s + 0.742·29-s + 0.898·31-s + 0.676·35-s + 1.47·37-s − 0.312·41-s + 0.914·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 0.650·59-s + 0.248·65-s + 1.58·67-s + 0.118·71-s + 1.63·73-s + 0.455·77-s − 1.12·79-s + 1.53·83-s + 1.37·89-s + 0.838·91-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 2.9504467442.950446744
L(12)L(\frac12) ≈\approx 2.9504467442.950446744
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−T+pT2 1 - 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+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+9T+pT2 1 + 9 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+5T+pT2 1 + 5 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−13T+pT2 1 - 13 T + p T^{2}
71 1−T+pT2 1 - T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1−13T+pT2 1 - 13 T + p T^{2}
97 1+19T+pT2 1 + 19 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

−8.071377128053344859894122622868, −7.56519678805781130407848167694, −6.44172225692526725575978123475, −5.95276051706536103373758696818, −5.17985594858562856526380380343, −4.40058804882834120573816231371, −3.81710770374657855777982606596, −2.54475665197812259665515911745, −1.81621868705930417076350207872, −0.955095243892024995431714781385, 0.955095243892024995431714781385, 1.81621868705930417076350207872, 2.54475665197812259665515911745, 3.81710770374657855777982606596, 4.40058804882834120573816231371, 5.17985594858562856526380380343, 5.95276051706536103373758696818, 6.44172225692526725575978123475, 7.56519678805781130407848167694, 8.071377128053344859894122622868

Graph of the ZZ-function along the critical line