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Properties

Label 4-41472-1.1-c1e2-0-6
Degree 44
Conductor 4147241472
Sign −1-1
Analytic cond. 2.644292.64429
Root an. cond. 1.275191.27519
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  − 8·11-s − 4·13-s − 8·23-s − 2·25-s − 4·37-s + 8·47-s + 2·49-s − 4·61-s − 8·71-s + 12·73-s − 8·83-s + 12·97-s − 16·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2.41·11-s − 1.10·13-s − 1.66·23-s − 2/5·25-s − 0.657·37-s + 1.16·47-s + 2/7·49-s − 0.512·61-s − 0.949·71-s + 1.40·73-s − 0.878·83-s + 1.21·97-s − 1.54·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=(41472s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(41472s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4147241472    =    29⋅342^{9} \cdot 3^{4}
Sign: −1-1
Analytic conductor: 2.644292.64429
Root analytic conductor: 1.275191.27519
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 41472, ( :1/2,1/2), −1)(4,\ 41472,\ (\ :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
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
43C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−14T+pT2)(1+2T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−14T+pT2)(1+2T+pT2) ( 1 - 14 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

−10.04239490327039020804834100895, −9.671137631678653310194703102301, −8.947220995210199516220363970479, −8.235856561928500880487725685498, −7.919149368288401056596003007042, −7.43954650029933091969931577343, −6.97479060152944198107820505538, −5.96645258709776056472230677282, −5.63174825127725680977095102702, −4.98232615845313737476367504219, −4.45409902508691909602462330470, −3.52353181965425220918934352536, −2.60736399031644305889760904369, −2.13395303201845493039367507448, 0, 2.13395303201845493039367507448, 2.60736399031644305889760904369, 3.52353181965425220918934352536, 4.45409902508691909602462330470, 4.98232615845313737476367504219, 5.63174825127725680977095102702, 5.96645258709776056472230677282, 6.97479060152944198107820505538, 7.43954650029933091969931577343, 7.919149368288401056596003007042, 8.235856561928500880487725685498, 8.947220995210199516220363970479, 9.671137631678653310194703102301, 10.04239490327039020804834100895

Graph of the ZZ-function along the critical line