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Properties

Label 4-476e2-1.1-c1e2-0-20
Degree 44
Conductor 226576226576
Sign 11
Analytic cond. 14.446614.4466
Root an. cond. 1.949581.94958
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s + 2·7-s + 4·9-s + 2·11-s − 6·13-s + 9·15-s − 2·17-s − 12·19-s − 6·21-s + 2·23-s − 6·27-s − 10·29-s − 17·31-s − 6·33-s − 6·35-s + 18·39-s + 41-s − 5·43-s − 12·45-s + 20·47-s + 3·49-s + 6·51-s + 5·53-s − 6·55-s + 36·57-s − 4·59-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s + 0.755·7-s + 4/3·9-s + 0.603·11-s − 1.66·13-s + 2.32·15-s − 0.485·17-s − 2.75·19-s − 1.30·21-s + 0.417·23-s − 1.15·27-s − 1.85·29-s − 3.05·31-s − 1.04·33-s − 1.01·35-s + 2.88·39-s + 0.156·41-s − 0.762·43-s − 1.78·45-s + 2.91·47-s + 3/7·49-s + 0.840·51-s + 0.686·53-s − 0.809·55-s + 4.76·57-s − 0.520·59-s + ⋯

Functional equation

Λ(s)=(226576s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(226576s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 226576226576    =    24⋅72⋅1722^{4} \cdot 7^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 14.446614.4466
Root analytic conductor: 1.949581.94958
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 226576, ( :1/2,1/2), 1)(4,\ 226576,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7C1C_1 (1−T)2 ( 1 - T )^{2}
17C1C_1 (1+T)2 ( 1 + T )^{2}
good3C4C_4 1+pT+5T2+p2T3+p2T4 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4}
5D4D_{4} 1+3T+9T2+3pT3+p2T4 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+6T+22T2+6pT3+p2T4 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23D4D_{4} 1−2T+34T2−2pT3+p2T4 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+10T+70T2+10pT3+p2T4 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+17T+131T2+17pT3+p2T4 1 + 17 T + 131 T^{2} + 17 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
41D4D_{4} 1−T+T2−pT3+p2T4 1 - T + T^{2} - p T^{3} + p^{2} T^{4}
43C4C_4 1+5T+89T2+5pT3+p2T4 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4}
47C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
53D4D_{4} 1−5T+109T2−5pT3+p2T4 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+4T+70T2+4pT3+p2T4 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+5T+125T2+5pT3+p2T4 1 + 5 T + 125 T^{2} + 5 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−9T+151T2−9pT3+p2T4 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4}
71C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
73D4D_{4} 1+9T+85T2+9pT3+p2T4 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4}
79C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
83D4D_{4} 1−4T+118T2−4pT3+p2T4 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97D4D_{4} 1+15T+221T2+15pT3+p2T4 1 + 15 T + 221 T^{2} + 15 p T^{3} + p^{2} T^{4}
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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

−10.85696882384837336755774984979, −10.79420332340129358871089145272, −10.08964048887404750722789904321, −9.440023554094045298409228756394, −8.838366900915876296429655943828, −8.739793279493932851068109336666, −7.76481959306128598922806051066, −7.50073954977690814783436592085, −7.15387354484516515755702646157, −6.70385753197726502977657401306, −5.85128632622433463351478609513, −5.69114875933762450620973709082, −5.09277283063707884389548729149, −4.47554244256704604327094464595, −4.01621350754607537551052472636, −3.80293690894226375210320066440, −2.32822010793954904243592174695, −1.80826892973584175146586308601, 0, 0, 1.80826892973584175146586308601, 2.32822010793954904243592174695, 3.80293690894226375210320066440, 4.01621350754607537551052472636, 4.47554244256704604327094464595, 5.09277283063707884389548729149, 5.69114875933762450620973709082, 5.85128632622433463351478609513, 6.70385753197726502977657401306, 7.15387354484516515755702646157, 7.50073954977690814783436592085, 7.76481959306128598922806051066, 8.739793279493932851068109336666, 8.838366900915876296429655943828, 9.440023554094045298409228756394, 10.08964048887404750722789904321, 10.79420332340129358871089145272, 10.85696882384837336755774984979

Graph of the ZZ-function along the critical line