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Properties

Label 4-888e2-1.1-c1e2-0-41
Degree 44
Conductor 788544788544
Sign 11
Analytic cond. 50.278250.2782
Root an. cond. 2.662832.66283
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s − 2·9-s − 12·13-s − 16·19-s − 2·21-s − 6·25-s + 5·27-s − 8·31-s − 2·37-s + 12·39-s + 4·43-s − 11·49-s + 16·57-s + 8·61-s − 4·63-s + 14·73-s + 6·75-s + 81-s − 24·91-s + 8·93-s − 16·97-s + 4·103-s + 32·109-s + 2·111-s + 24·117-s − 21·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 2/3·9-s − 3.32·13-s − 3.67·19-s − 0.436·21-s − 6/5·25-s + 0.962·27-s − 1.43·31-s − 0.328·37-s + 1.92·39-s + 0.609·43-s − 1.57·49-s + 2.11·57-s + 1.02·61-s − 0.503·63-s + 1.63·73-s + 0.692·75-s + 1/9·81-s − 2.51·91-s + 0.829·93-s − 1.62·97-s + 0.394·103-s + 3.06·109-s + 0.189·111-s + 2.21·117-s − 1.90·121-s + ⋯

Functional equation

Λ(s)=(788544s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(788544s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 788544788544    =    26⋅32⋅3722^{6} \cdot 3^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 50.278250.2782
Root analytic conductor: 2.662832.66283
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 788544, ( :1/2,1/2), 1)(4,\ 788544,\ (\ :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+T+pT2 1 + T + p T^{2}
37C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
17C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
43C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
47C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
73C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
89C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
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

−7.78790307165276969482845597819, −7.36881114003389544173968798730, −6.93610596286674052907325458245, −6.40622823243653800847643763088, −6.09037614955138614512431406118, −5.35667365720479356016286420004, −5.06148787959059824431190870012, −4.67375669110578511814790030096, −4.21402845745683233262500412192, −3.62376221698954049890975125760, −2.50673634667839885265978774188, −2.32752817979750411282892328426, −1.85774356352190049046929099208, 0, 0, 1.85774356352190049046929099208, 2.32752817979750411282892328426, 2.50673634667839885265978774188, 3.62376221698954049890975125760, 4.21402845745683233262500412192, 4.67375669110578511814790030096, 5.06148787959059824431190870012, 5.35667365720479356016286420004, 6.09037614955138614512431406118, 6.40622823243653800847643763088, 6.93610596286674052907325458245, 7.36881114003389544173968798730, 7.78790307165276969482845597819

Graph of the ZZ-function along the critical line