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Properties

Label 4-312e2-1.1-c1e2-0-46
Degree 44
Conductor 9734497344
Sign −1-1
Analytic cond. 6.206736.20673
Root an. cond. 1.578391.57839
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 2·13-s − 12·17-s − 10·25-s − 12·29-s + 4·37-s − 24·41-s − 10·49-s + 12·53-s + 4·61-s + 28·73-s + 81-s − 20·97-s + 36·101-s + 28·109-s + 12·113-s + 2·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 0.554·13-s − 2.91·17-s − 2·25-s − 2.22·29-s + 0.657·37-s − 3.74·41-s − 1.42·49-s + 1.64·53-s + 0.512·61-s + 3.27·73-s + 1/9·81-s − 2.03·97-s + 3.58·101-s + 2.68·109-s + 1.12·113-s + 0.184·117-s − 2·121-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 6.206736.20673
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 97344, ( :1/2,1/2), −1)(4,\ 97344,\ (\ :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
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
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}
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−9.385896632264966978634773161619, −8.735814680045936069504206621070, −8.531128945035369322451694692338, −7.86871640882441783507560416572, −7.32625330559643707687229250500, −6.65843891177652373785895186115, −6.49558938607564697804430447032, −5.75513068077913273540546960045, −5.11454570698520102977754794044, −4.55277073254635650297368891844, −3.80067854107620357152276821608, −3.54130855144350797063436114136, −2.03881758362108754018941589683, −2.01665432147547119953995139615, 0, 2.01665432147547119953995139615, 2.03881758362108754018941589683, 3.54130855144350797063436114136, 3.80067854107620357152276821608, 4.55277073254635650297368891844, 5.11454570698520102977754794044, 5.75513068077913273540546960045, 6.49558938607564697804430447032, 6.65843891177652373785895186115, 7.32625330559643707687229250500, 7.86871640882441783507560416572, 8.531128945035369322451694692338, 8.735814680045936069504206621070, 9.385896632264966978634773161619

Graph of the ZZ-function along the critical line