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Properties

Label 4-1596e2-1.1-c1e2-0-25
Degree 44
Conductor 25472162547216
Sign 11
Analytic cond. 162.412162.412
Root an. cond. 3.569893.56989
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 2·7-s + 3·9-s − 8·13-s − 4·15-s − 2·17-s − 2·19-s − 4·21-s − 8·23-s − 4·25-s + 4·27-s − 2·29-s − 4·31-s + 4·35-s − 8·37-s − 16·39-s − 12·41-s − 6·45-s + 2·47-s + 3·49-s − 4·51-s + 14·53-s − 4·57-s − 8·59-s − 8·61-s − 6·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 2.21·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 4/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 1.31·37-s − 2.56·39-s − 1.87·41-s − 0.894·45-s + 0.291·47-s + 3/7·49-s − 0.560·51-s + 1.92·53-s − 0.529·57-s − 1.04·59-s − 1.02·61-s − 0.755·63-s + ⋯

Functional equation

Λ(s)=(2547216s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2547216s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25472162547216    =    24⋅32⋅72⋅1922^{4} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 162.412162.412
Root analytic conductor: 3.569893.56989
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2547216, ( :1/2,1/2), 1)(4,\ 2547216,\ (\ :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 (1−T)2 ( 1 - T )^{2}
7C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good5D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+8T+50T2+8pT3+p2T4 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+32T2+2pT3+p2T4 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+8T+78T2+8pT3+p2T4 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+12T+106T2+12pT3+p2T4 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
47D4D_{4} 1−2T+92T2−2pT3+p2T4 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−14T+152T2−14pT3+p2T4 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+8T+86T2+8pT3+p2T4 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+4T+126T2+4pT3+p2T4 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−10T+140T2−10pT3+p2T4 1 - 10 T + 140 T^{2} - 10 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+4T−42T2+4pT3+p2T4 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+20T+246T2+20pT3+p2T4 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+6T+172T2+6pT3+p2T4 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−12T+202T2−12pT3+p2T4 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−9.049756608322081150738194287119, −8.937904194160606182488496591659, −8.269437357771425466382722793017, −8.151184128319022201781218678710, −7.50335456242339517333425498000, −7.40704885167260961065635383357, −6.85319567342577352402456201956, −6.75756065243963087479864872555, −5.77389054622750536507354432462, −5.68748622098255744291745446945, −4.83145021884166292200148545044, −4.55381018080794315093605473201, −3.87618389872982774635788888351, −3.81769987381258158511808038358, −3.12755645889295944342395056228, −2.65895010302699426090860535190, −2.06685566834651565159805977056, −1.74629627466307893717266668862, 0, 0, 1.74629627466307893717266668862, 2.06685566834651565159805977056, 2.65895010302699426090860535190, 3.12755645889295944342395056228, 3.81769987381258158511808038358, 3.87618389872982774635788888351, 4.55381018080794315093605473201, 4.83145021884166292200148545044, 5.68748622098255744291745446945, 5.77389054622750536507354432462, 6.75756065243963087479864872555, 6.85319567342577352402456201956, 7.40704885167260961065635383357, 7.50335456242339517333425498000, 8.151184128319022201781218678710, 8.269437357771425466382722793017, 8.937904194160606182488496591659, 9.049756608322081150738194287119

Graph of the ZZ-function along the critical line