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Properties

Label 4-972e2-1.1-c1e2-0-1
Degree 44
Conductor 944784944784
Sign 11
Analytic cond. 60.240260.2402
Root an. cond. 2.785932.78593
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 5·13-s − 14·19-s + 5·25-s + 7·31-s − 20·37-s + 13·43-s + 7·49-s + 13·61-s − 11·67-s + 34·73-s − 17·79-s − 20·91-s + 19·97-s + 7·103-s + 34·109-s + 11·121-s + 127-s + 131-s − 56·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.38·13-s − 3.21·19-s + 25-s + 1.25·31-s − 3.28·37-s + 1.98·43-s + 49-s + 1.66·61-s − 1.34·67-s + 3.97·73-s − 1.91·79-s − 2.09·91-s + 1.92·97-s + 0.689·103-s + 3.25·109-s + 121-s + 0.0887·127-s + 0.0873·131-s − 4.85·133-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)=(944784s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(944784s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 944784944784    =    24⋅3102^{4} \cdot 3^{10}
Sign: 11
Analytic conductor: 60.240260.2402
Root analytic conductor: 2.785932.78593
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 944784, ( :1/2,1/2), 1)(4,\ 944784,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7863337151.786333715
L(12)L(\frac12) ≈\approx 1.7863337151.786333715
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
good5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1−11T+pT2)(1+4T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1−5T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} )
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C2C_2 (1−14T+pT2)(1+T+pT2) ( 1 - 14 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 (1−17T+pT2)2 ( 1 - 17 T + p T^{2} )^{2}
79C2C_2 (1+4T+pT2)(1+13T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1−5T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 5 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

−10.22462097453571218112343460634, −10.18697061169295571038824450105, −9.162377778935506121976372507627, −8.756652611867347707245320640521, −8.739823279221469597398495088759, −8.124202131819958128206303762430, −7.83153989491991082327661306619, −7.27313939997081265463475314151, −6.80148584373458156962987207243, −6.53627614476559093084639588800, −5.93437472438920757150385202680, −5.23741506028052292590218272383, −4.97173078420817295032312813270, −4.51043623003399902229115609309, −4.19237772252609825951327090288, −3.50557625438951798957335859084, −2.62032001267920941684449414407, −2.12889956551082042608832160227, −1.81224222374119512359537461234, −0.60557878200296286027696848784, 0.60557878200296286027696848784, 1.81224222374119512359537461234, 2.12889956551082042608832160227, 2.62032001267920941684449414407, 3.50557625438951798957335859084, 4.19237772252609825951327090288, 4.51043623003399902229115609309, 4.97173078420817295032312813270, 5.23741506028052292590218272383, 5.93437472438920757150385202680, 6.53627614476559093084639588800, 6.80148584373458156962987207243, 7.27313939997081265463475314151, 7.83153989491991082327661306619, 8.124202131819958128206303762430, 8.739823279221469597398495088759, 8.756652611867347707245320640521, 9.162377778935506121976372507627, 10.18697061169295571038824450105, 10.22462097453571218112343460634

Graph of the ZZ-function along the critical line