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Properties

Label 4-936e2-1.1-c1e2-0-19
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 4·13-s − 4·19-s − 4·25-s + 2·37-s + 12·43-s + 2·49-s − 8·61-s + 8·67-s + 6·73-s + 8·79-s + 16·91-s + 14·97-s − 20·103-s + 6·109-s + 8·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.10·13-s − 0.917·19-s − 4/5·25-s + 0.328·37-s + 1.82·43-s + 2/7·49-s − 1.02·61-s + 0.977·67-s + 0.702·73-s + 0.900·79-s + 1.67·91-s + 1.42·97-s − 1.97·103-s + 0.574·109-s + 8/11·121-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 + 3/13·169-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: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5876319922.587631992
L(12)L(\frac12) ≈\approx 2.5876319922.587631992
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
13C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good5C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
11C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
17C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+8T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1+52T2+p2T4 1 + 52 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−8T+pT2)(1−4T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )
47C22C_2^2 1−80T2+p2T4 1 - 80 T^{2} + p^{2} T^{4}
53C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
59C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1−128T2+p2T4 1 - 128 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+4T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−12T+pT2)(1+4T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−56T2+p2T4 1 - 56 T^{2} + p^{2} T^{4}
89C22C_2^2 1+60T2+p2T4 1 + 60 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−12T+pT2)(1−2T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 2 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

−8.117965845767349000385144246676, −7.916862978599173827885363508038, −7.45124827435846047304532781649, −6.87233367513434423874764469745, −6.32084908146833985179775165536, −5.99510368810614965500153829853, −5.49025366834376216054893233753, −4.98721499661367299522041010263, −4.45761727249049658805622833757, −4.08496261187088214791451424382, −3.59571463463586332183810724353, −2.81067293210197952086372659559, −2.07862773707338337245971139521, −1.65519517675676495026349981393, −0.799092026316408552285567257007, 0.799092026316408552285567257007, 1.65519517675676495026349981393, 2.07862773707338337245971139521, 2.81067293210197952086372659559, 3.59571463463586332183810724353, 4.08496261187088214791451424382, 4.45761727249049658805622833757, 4.98721499661367299522041010263, 5.49025366834376216054893233753, 5.99510368810614965500153829853, 6.32084908146833985179775165536, 6.87233367513434423874764469745, 7.45124827435846047304532781649, 7.916862978599173827885363508038, 8.117965845767349000385144246676

Graph of the ZZ-function along the critical line