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Properties

Label 4-936e2-1.1-c1e2-0-41
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
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 + 6·9-s − 6·11-s − 5·13-s + 9·15-s + 5·17-s + 3·19-s − 8·23-s + 5·25-s − 9·27-s − 18·29-s + 7·31-s + 18·33-s + 5·37-s + 15·39-s − 6·41-s + 8·43-s − 18·45-s + 47-s + 7·49-s − 15·51-s + 20·53-s + 18·55-s − 9·57-s − 18·59-s + 6·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s + 2·9-s − 1.80·11-s − 1.38·13-s + 2.32·15-s + 1.21·17-s + 0.688·19-s − 1.66·23-s + 25-s − 1.73·27-s − 3.34·29-s + 1.25·31-s + 3.13·33-s + 0.821·37-s + 2.40·39-s − 0.937·41-s + 1.21·43-s − 2.68·45-s + 0.145·47-s + 49-s − 2.10·51-s + 2.74·53-s + 2.42·55-s − 1.19·57-s − 2.34·59-s + 0.768·61-s + ⋯

Functional equation

Λ(s)=(876096s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(876096s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26⋅34⋅1322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :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
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
7C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
17C22C_2^2 1−5T+8T2−5pT3+p2T4 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−3T−10T2−3pT3+p2T4 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}
29C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+4T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C22C_2^2 1−5T−12T2−5pT3+p2T4 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+6T−5T2+6pT3+p2T4 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4}
43C2C_2 (1−13T+pT2)(1+5T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} )
47C22C_2^2 1−T−46T2−pT3+p2T4 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
61C22C_2^2 1−6T−25T2−6pT3+p2T4 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+12T+77T2+12pT3+p2T4 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+T−70T2+pT3+p2T4 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79C22C_2^2 1+11T+42T2+11pT3+p2T4 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+3T−74T2+3pT3+p2T4 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+11T+32T2+11pT3+p2T4 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4}
97C2C_2 (1−5T+pT2)(1+19T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{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

−10.21588692221203179500467666783, −9.616774568013260844088834770050, −9.157743022436732513407789051753, −8.434824960489590997617174678118, −7.83452805084155089333762493145, −7.62856631743935795062380073440, −7.30418638627036425691365154569, −7.22625770743461330208687199847, −6.26223092524026538078556517028, −5.63002330909251917615298605546, −5.58050928132307884244095569482, −5.25104962021962065413101676039, −4.38664074109845751692807810484, −4.30958139307583089520083666362, −3.64115797832701012179039388111, −2.86962857196961273102971430762, −2.28803540315174946141778946955, −1.22295559690610853878416816173, 0, 0, 1.22295559690610853878416816173, 2.28803540315174946141778946955, 2.86962857196961273102971430762, 3.64115797832701012179039388111, 4.30958139307583089520083666362, 4.38664074109845751692807810484, 5.25104962021962065413101676039, 5.58050928132307884244095569482, 5.63002330909251917615298605546, 6.26223092524026538078556517028, 7.22625770743461330208687199847, 7.30418638627036425691365154569, 7.62856631743935795062380073440, 7.83452805084155089333762493145, 8.434824960489590997617174678118, 9.157743022436732513407789051753, 9.616774568013260844088834770050, 10.21588692221203179500467666783

Graph of the ZZ-function along the critical line