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Properties

Label 4-936e2-1.1-c1e2-0-47
Degree 44
Conductor 876096876096
Sign −1-1
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 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 2·13-s − 4·19-s − 6·25-s + 4·31-s − 20·37-s + 16·43-s − 2·49-s − 12·61-s − 12·67-s − 4·73-s + 24·79-s − 8·91-s − 20·97-s + 8·103-s + 20·109-s − 6·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.554·13-s − 0.917·19-s − 6/5·25-s + 0.718·31-s − 3.28·37-s + 2.43·43-s − 2/7·49-s − 1.53·61-s − 1.46·67-s − 0.468·73-s + 2.70·79-s − 0.838·91-s − 2.03·97-s + 0.788·103-s + 1.91·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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: −1-1
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: 11
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
3 1 1
13C1C_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 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 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+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
67C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−7.917819582253129208914516956622, −7.44699838933626723226640796085, −7.40568873395535407748018257450, −6.61084334022363680034133660955, −6.16078101072896661105492972029, −5.73335318755003699286776095361, −5.08169261977535984530512865091, −4.82890466759038882574961362490, −4.35999893307754658375735542809, −3.80548435074655755032944611187, −3.20854141442725326453087823255, −2.36222722805652747794129158747, −1.93418903533460519772261561653, −1.32780807021178389715142739482, 0, 1.32780807021178389715142739482, 1.93418903533460519772261561653, 2.36222722805652747794129158747, 3.20854141442725326453087823255, 3.80548435074655755032944611187, 4.35999893307754658375735542809, 4.82890466759038882574961362490, 5.08169261977535984530512865091, 5.73335318755003699286776095361, 6.16078101072896661105492972029, 6.61084334022363680034133660955, 7.40568873395535407748018257450, 7.44699838933626723226640796085, 7.917819582253129208914516956622

Graph of the ZZ-function along the critical line