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Properties

Label 4-345600-1.1-c1e2-0-12
Degree 44
Conductor 345600345600
Sign 11
Analytic cond. 22.035722.0357
Root an. cond. 2.166612.16661
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s + 2·15-s − 8·19-s + 3·25-s + 27-s − 4·29-s + 24·43-s + 2·45-s + 16·47-s − 14·49-s + 12·53-s − 8·57-s + 8·67-s + 16·71-s − 12·73-s + 3·75-s + 81-s − 4·87-s − 16·95-s + 4·97-s + 12·101-s − 6·121-s + 4·125-s + 127-s + 24·129-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s − 1.83·19-s + 3/5·25-s + 0.192·27-s − 0.742·29-s + 3.65·43-s + 0.298·45-s + 2.33·47-s − 2·49-s + 1.64·53-s − 1.05·57-s + 0.977·67-s + 1.89·71-s − 1.40·73-s + 0.346·75-s + 1/9·81-s − 0.428·87-s − 1.64·95-s + 0.406·97-s + 1.19·101-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 2.11·129-s + ⋯

Functional equation

Λ(s)=(345600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(345600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345600345600    =    29⋅33⋅522^{9} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 22.035722.0357
Root analytic conductor: 2.166612.16661
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 345600, ( :1/2,1/2), 1)(4,\ 345600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6317574532.631757453
L(12)L(\frac12) ≈\approx 2.6317574532.631757453
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 1 - T
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + 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 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 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 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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

−8.979673560126252996904408641448, −8.323057177226078640880314002796, −7.85971332456036455091052470017, −7.36860587905514204016946354425, −6.89270810769313104844169224865, −6.38799072784883331024834438481, −5.83633635666208002119272737589, −5.59585158632682227968586342616, −4.81681980421746127667781868965, −4.14224381134792067626093257234, −3.95303834850677407956821293662, −3.00246618464052305255387261031, −2.25002432004047770108316173263, −2.12011231568422406855465196308, −0.910745729015585711280898694007, 0.910745729015585711280898694007, 2.12011231568422406855465196308, 2.25002432004047770108316173263, 3.00246618464052305255387261031, 3.95303834850677407956821293662, 4.14224381134792067626093257234, 4.81681980421746127667781868965, 5.59585158632682227968586342616, 5.83633635666208002119272737589, 6.38799072784883331024834438481, 6.89270810769313104844169224865, 7.36860587905514204016946354425, 7.85971332456036455091052470017, 8.323057177226078640880314002796, 8.979673560126252996904408641448

Graph of the ZZ-function along the critical line