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Properties

Label 4-1216e2-1.1-c1e2-0-28
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
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  − 5·9-s − 2·13-s − 10·17-s − 10·25-s + 6·29-s − 4·37-s − 16·41-s − 5·49-s − 18·53-s − 28·61-s + 18·73-s + 16·81-s − 24·89-s + 28·97-s + 28·101-s − 14·109-s − 36·113-s + 10·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 50·153-s + 157-s + ⋯
L(s)  = 1  − 5/3·9-s − 0.554·13-s − 2.42·17-s − 2·25-s + 1.11·29-s − 0.657·37-s − 2.49·41-s − 5/7·49-s − 2.47·53-s − 3.58·61-s + 2.10·73-s + 16/9·81-s − 2.54·89-s + 2.84·97-s + 2.78·101-s − 1.34·109-s − 3.38·113-s + 0.924·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 4.04·153-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
17C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
23C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
59C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C2C_2 (1−13T+pT2)(1+13T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)2 ( 1 - 14 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

−7.74515606034087698025852781244, −6.73715959969007181198215210739, −6.72377104864689556277323751664, −6.17043864352631516446211866530, −5.90719710332910393553929521875, −5.03277316009204668285879882866, −4.99527268882455447172353841146, −4.41313931480294198592546395456, −3.81156163044135505245852369486, −3.09698588528238924660584466755, −2.89182953598080292522833763992, −2.01374174404058579498750722555, −1.78091799751097958894751684261, 0, 0, 1.78091799751097958894751684261, 2.01374174404058579498750722555, 2.89182953598080292522833763992, 3.09698588528238924660584466755, 3.81156163044135505245852369486, 4.41313931480294198592546395456, 4.99527268882455447172353841146, 5.03277316009204668285879882866, 5.90719710332910393553929521875, 6.17043864352631516446211866530, 6.72377104864689556277323751664, 6.73715959969007181198215210739, 7.74515606034087698025852781244

Graph of the ZZ-function along the critical line