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Properties

Label 4-240e2-1.1-c1e2-0-24
Degree 44
Conductor 5760057600
Sign −1-1
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s − 7·19-s − 5·25-s − 9·27-s + 13·49-s + 21·57-s − 21·67-s − 24·73-s + 15·75-s + 9·81-s − 11·121-s + 127-s + 131-s + 137-s + 139-s − 39·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s − 42·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 1.60·19-s − 25-s − 1.73·27-s + 13/7·49-s + 2.78·57-s − 2.56·67-s − 2.80·73-s + 1.73·75-s + 81-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.21·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s − 3.21·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(57600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(57600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5760057600    =    28⋅32⋅522^{8} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 57600, ( :1/2,1/2), −1)(4,\ 57600,\ (\ :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}
5C2C_2 1+pT2 1 + p T^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
13C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C2C_2 (1−T+pT2)(1+8T+pT2) ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
43C2C_2 (1−13T+pT2)(1+13T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
67C2C_2 (1+5T+pT2)(1+16T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+7T+pT2)(1+17T+pT2) ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
89C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+19T+pT2) ( 1 - 19 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

−10.03289735748629283952937134769, −9.270981567679906681995908952508, −8.784774390971334014644934694365, −8.189478920900444648462771977182, −7.36990395992700175941088900995, −7.14839517916630981371693058846, −6.37808766560145263834313988983, −5.90629154350294878239687671104, −5.69678869113846324851276172303, −4.74937257504541601053316791273, −4.39738860790570192073165449065, −3.74799988141654562579655820524, −2.51332371484229190745930153240, −1.47089399278593014390941848189, 0, 1.47089399278593014390941848189, 2.51332371484229190745930153240, 3.74799988141654562579655820524, 4.39738860790570192073165449065, 4.74937257504541601053316791273, 5.69678869113846324851276172303, 5.90629154350294878239687671104, 6.37808766560145263834313988983, 7.14839517916630981371693058846, 7.36990395992700175941088900995, 8.189478920900444648462771977182, 8.784774390971334014644934694365, 9.270981567679906681995908952508, 10.03289735748629283952937134769

Graph of the ZZ-function along the critical line