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Properties

Label 4-810e2-1.1-c1e2-0-8
Degree 44
Conductor 656100656100
Sign 11
Analytic cond. 41.833541.8335
Root an. cond. 2.543202.54320
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  − 4-s + 2·5-s + 4·11-s + 16-s − 12·19-s − 2·20-s − 25-s + 18·29-s − 4·31-s + 22·41-s − 4·44-s + 13·49-s + 8·55-s − 8·59-s − 14·61-s − 64-s + 12·71-s + 12·76-s + 24·79-s + 2·80-s + 2·89-s − 24·95-s + 100-s − 4·101-s − 14·109-s − 18·116-s − 10·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.20·11-s + 1/4·16-s − 2.75·19-s − 0.447·20-s − 1/5·25-s + 3.34·29-s − 0.718·31-s + 3.43·41-s − 0.603·44-s + 13/7·49-s + 1.07·55-s − 1.04·59-s − 1.79·61-s − 1/8·64-s + 1.42·71-s + 1.37·76-s + 2.70·79-s + 0.223·80-s + 0.211·89-s − 2.46·95-s + 1/10·100-s − 0.398·101-s − 1.34·109-s − 1.67·116-s − 0.909·121-s + ⋯

Functional equation

Λ(s)=(656100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(656100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 656100656100    =    22⋅38⋅522^{2} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 41.833541.8335
Root analytic conductor: 2.543202.54320
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 656100, ( :1/2,1/2), 1)(4,\ 656100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1549092722.154909272
L(12)L(\frac12) ≈\approx 2.1549092722.154909272
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)
bad2C2C_2 1+T2 1 + T^{2}
3 1 1
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
29C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
67C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
89C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−10.52197178555317896387040196744, −10.04800974545498629268388419157, −9.514610391712751301862732595419, −9.081051582027114513545492207488, −8.953718065377669895966000325791, −8.529198657071138783898618119802, −7.77448706034300682021961158555, −7.75830231638378720781233462230, −6.62177574132703477385759901646, −6.55531831013259950064366906937, −6.20225897781143317383225486233, −5.78117883331226330388537252732, −5.02367800095249948598548689840, −4.56947758903200268615128018521, −4.03473183604154958924012162300, −3.88574082540073227006508084181, −2.60642453590614127625623953694, −2.51285276101068256897793327535, −1.57985110452461566651549498578, −0.77682335440868122790515043342, 0.77682335440868122790515043342, 1.57985110452461566651549498578, 2.51285276101068256897793327535, 2.60642453590614127625623953694, 3.88574082540073227006508084181, 4.03473183604154958924012162300, 4.56947758903200268615128018521, 5.02367800095249948598548689840, 5.78117883331226330388537252732, 6.20225897781143317383225486233, 6.55531831013259950064366906937, 6.62177574132703477385759901646, 7.75830231638378720781233462230, 7.77448706034300682021961158555, 8.529198657071138783898618119802, 8.953718065377669895966000325791, 9.081051582027114513545492207488, 9.514610391712751301862732595419, 10.04800974545498629268388419157, 10.52197178555317896387040196744

Graph of the ZZ-function along the critical line