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Properties

Label 4-41472-1.1-c1e2-0-4
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  − 4·5-s + 6·25-s − 4·29-s − 16·43-s − 16·47-s + 2·49-s − 4·53-s − 16·67-s − 4·73-s − 4·97-s + 12·101-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.78·5-s + 6/5·25-s − 0.742·29-s − 2.43·43-s − 2.33·47-s + 2/7·49-s − 0.549·53-s − 1.95·67-s − 0.468·73-s − 0.406·97-s + 1.19·101-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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
good5C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1−114T2+p2T4 1 - 114 T^{2} + p^{2} T^{4}
83C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
89C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C2C_2 (1+2T+pT2)2 ( 1 + 2 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

−10.04046016536150391071226454197, −9.478769385172889937032838588050, −8.822269054713076576399042446863, −8.277923334369285600815885459629, −7.965501018749938574846737817160, −7.42983512048804019983419743280, −6.90753797982818291117957608583, −6.33118006148456070158803503814, −5.52630893917906434361953259655, −4.75595304641916474324629799425, −4.33747103184689078316541533761, −3.47249024434556441041760301270, −3.17776636312574109281846582860, −1.74782644869831230888825208996, 0, 1.74782644869831230888825208996, 3.17776636312574109281846582860, 3.47249024434556441041760301270, 4.33747103184689078316541533761, 4.75595304641916474324629799425, 5.52630893917906434361953259655, 6.33118006148456070158803503814, 6.90753797982818291117957608583, 7.42983512048804019983419743280, 7.965501018749938574846737817160, 8.277923334369285600815885459629, 8.822269054713076576399042446863, 9.478769385172889937032838588050, 10.04046016536150391071226454197

Graph of the ZZ-function along the critical line