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Properties

Label 4-987696-1.1-c1e2-0-9
Degree 44
Conductor 987696987696
Sign 11
Analytic cond. 62.976362.9763
Root an. cond. 2.817042.81704
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 7·5-s + 3·8-s − 3·9-s + 7·10-s − 16-s − 11·17-s + 3·18-s + 19-s + 7·20-s + 27·25-s + 9·31-s − 5·32-s + 11·34-s + 3·36-s − 38-s − 21·40-s + 21·45-s − 5·49-s − 27·50-s + 8·59-s + 9·61-s − 9·62-s + 7·64-s − 25·67-s + 11·68-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 3.13·5-s + 1.06·8-s − 9-s + 2.21·10-s − 1/4·16-s − 2.66·17-s + 0.707·18-s + 0.229·19-s + 1.56·20-s + 27/5·25-s + 1.61·31-s − 0.883·32-s + 1.88·34-s + 1/2·36-s − 0.162·38-s − 3.32·40-s + 3.13·45-s − 5/7·49-s − 3.81·50-s + 1.04·59-s + 1.15·61-s − 1.14·62-s + 7/8·64-s − 3.05·67-s + 1.33·68-s + ⋯

Functional equation

Λ(s)=(987696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(987696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 987696987696    =    24⋅32⋅1932^{4} \cdot 3^{2} \cdot 19^{3}
Sign: 11
Analytic conductor: 62.976362.9763
Root analytic conductor: 2.817042.81704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 987696, ( :1/2,1/2), 1)(4,\ 987696,\ (\ :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)
bad2C2C_2 1+T+pT2 1 + T + p T^{2}
3C2C_2 1+pT2 1 + p T^{2}
19C1C_1 1−T 1 - T
good5C2C_2×\timesC2C_2 (1+3T+pT2)(1+4T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1+4T+pT2)(1+7T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
29C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−5T+pT2)(1−4T+pT2) ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} )
37C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
41C22C_2^2 1+17T2+p2T4 1 + 17 T^{2} + p^{2} T^{4}
43C22C_2^2 1−31T2+p2T4 1 - 31 T^{2} + p^{2} T^{4}
47C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
53C22C_2^2 1−101T2+p2T4 1 - 101 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−13T+pT2)(1+5T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−7T+pT2)(1−2T+pT2) ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} )
67C2C_2×\timesC2C_2 (1+11T+pT2)(1+14T+pT2) ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1+13T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−T+pT2)(1+7T+pT2) ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
89C22C_2^2 1−103T2+p2T4 1 - 103 T^{2} + p^{2} T^{4}
97C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + 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

−7.66940097960079654968343320759, −7.52528882656985284840224685193, −7.07576097962381727671007390031, −6.53385733939255667419175852884, −6.11934030024318110818189433734, −5.12892818867964033948783241084, −4.75188746992265904775058613465, −4.34326336593079571239619710955, −4.09178962006457974942008104448, −3.53767065607358261145384427024, −2.96679856343838499555264092530, −2.32469560550463480321015561966, −1.03533847979788543800629084484, 0, 0, 1.03533847979788543800629084484, 2.32469560550463480321015561966, 2.96679856343838499555264092530, 3.53767065607358261145384427024, 4.09178962006457974942008104448, 4.34326336593079571239619710955, 4.75188746992265904775058613465, 5.12892818867964033948783241084, 6.11934030024318110818189433734, 6.53385733939255667419175852884, 7.07576097962381727671007390031, 7.52528882656985284840224685193, 7.66940097960079654968343320759

Graph of the ZZ-function along the critical line