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Properties

Label 4-3344e2-1.1-c1e2-0-2
Degree 44
Conductor 1118233611182336
Sign 11
Analytic cond. 712.995712.995
Root an. cond. 5.167395.16739
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 5·7-s − 2·11-s + 7·13-s + 3·15-s − 6·17-s − 2·19-s + 5·21-s − 6·23-s + 2·25-s + 2·27-s + 9·29-s + 17·31-s − 2·33-s + 15·35-s + 16·37-s + 7·39-s + 3·41-s − 43-s + 10·49-s − 6·51-s + 6·53-s − 6·55-s − 2·57-s + 4·61-s + 21·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1.88·7-s − 0.603·11-s + 1.94·13-s + 0.774·15-s − 1.45·17-s − 0.458·19-s + 1.09·21-s − 1.25·23-s + 2/5·25-s + 0.384·27-s + 1.67·29-s + 3.05·31-s − 0.348·33-s + 2.53·35-s + 2.63·37-s + 1.12·39-s + 0.468·41-s − 0.152·43-s + 10/7·49-s − 0.840·51-s + 0.824·53-s − 0.809·55-s − 0.264·57-s + 0.512·61-s + 2.60·65-s + ⋯

Functional equation

Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11182336s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1118233611182336    =    28⋅112⋅1922^{8} \cdot 11^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 712.995712.995
Root analytic conductor: 5.167395.16739
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11182336, ( :1/2,1/2), 1)(4,\ 11182336,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.6688084747.668808474
L(12)L(\frac12) ≈\approx 7.6688084747.668808474
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1−T+T2−pT3+p2T4 1 - T + T^{2} - p T^{3} + p^{2} T^{4}
5D4D_{4} 1−3T+7T2−3pT3+p2T4 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4}
7D4D_{4} 1−5T+15T2−5pT3+p2T4 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−7T+33T2−7pT3+p2T4 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+6T+22T2+6pT3+p2T4 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+6T+34T2+6pT3+p2T4 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−9T+73T2−9pT3+p2T4 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−17T+129T2−17pT3+p2T4 1 - 17 T + 129 T^{2} - 17 p T^{3} + p^{2} T^{4}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41D4D_{4} 1−3T+79T2−3pT3+p2T4 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+T+39T2+pT3+p2T4 1 + T + 39 T^{2} + p T^{3} + p^{2} T^{4}
47C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
53D4D_{4} 1−6T+94T2−6pT3+p2T4 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67D4D_{4} 1−5T+135T2−5pT3+p2T4 1 - 5 T + 135 T^{2} - 5 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−9T+115T2−9pT3+p2T4 1 - 9 T + 115 T^{2} - 9 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−10T+150T2−10pT3+p2T4 1 - 10 T + 150 T^{2} - 10 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T+138T2−2pT3+p2T4 1 - 2 T + 138 T^{2} - 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−3T+163T2−3pT3+p2T4 1 - 3 T + 163 T^{2} - 3 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+18T+238T2+18pT3+p2T4 1 + 18 T + 238 T^{2} + 18 p T^{3} + p^{2} T^{4}
97C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.486807185866756808031225617644, −8.400399505010108897872101436719, −8.142490351811044314131412856339, −8.097714889068247609037348950572, −7.51981563924045934483629141209, −6.61833617184988910411403824790, −6.46654781869563031625894102157, −6.40355553839369254166287429616, −5.74906222594953493391441377177, −5.49619087486239944381125007344, −4.92906925361413416353411607223, −4.48925273948586296338615918432, −4.23917730801176553698487534301, −3.99105368593266749617840023358, −2.99004905331838308003568619485, −2.63289081188703848382747190153, −2.26052903083815587738649227849, −1.94377543123514766152516822837, −1.10605713476119079480259838507, −0.983137180192912163317366735551, 0.983137180192912163317366735551, 1.10605713476119079480259838507, 1.94377543123514766152516822837, 2.26052903083815587738649227849, 2.63289081188703848382747190153, 2.99004905331838308003568619485, 3.99105368593266749617840023358, 4.23917730801176553698487534301, 4.48925273948586296338615918432, 4.92906925361413416353411607223, 5.49619087486239944381125007344, 5.74906222594953493391441377177, 6.40355553839369254166287429616, 6.46654781869563031625894102157, 6.61833617184988910411403824790, 7.51981563924045934483629141209, 8.097714889068247609037348950572, 8.142490351811044314131412856339, 8.400399505010108897872101436719, 8.486807185866756808031225617644

Graph of the ZZ-function along the critical line