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Properties

Label 4-5400e2-1.1-c1e2-0-36
Degree 44
Conductor 2916000029160000
Sign 11
Analytic cond. 1859.261859.26
Root an. cond. 6.566526.56652
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·11-s + 2·13-s − 2·19-s + 4·23-s − 4·29-s + 12·31-s − 10·37-s − 4·41-s + 12·43-s − 11·49-s − 4·53-s − 2·61-s + 10·67-s − 28·71-s − 2·73-s + 8·77-s + 2·79-s + 20·83-s − 24·89-s − 4·91-s − 22·97-s − 24·101-s − 2·103-s − 4·107-s − 20·113-s − 4·121-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.20·11-s + 0.554·13-s − 0.458·19-s + 0.834·23-s − 0.742·29-s + 2.15·31-s − 1.64·37-s − 0.624·41-s + 1.82·43-s − 1.57·49-s − 0.549·53-s − 0.256·61-s + 1.22·67-s − 3.32·71-s − 0.234·73-s + 0.911·77-s + 0.225·79-s + 2.19·83-s − 2.54·89-s − 0.419·91-s − 2.23·97-s − 2.38·101-s − 0.197·103-s − 0.386·107-s − 1.88·113-s − 0.363·121-s + ⋯

Functional equation

Λ(s)=(29160000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29160000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2916000029160000    =    26⋅36⋅542^{6} \cdot 3^{6} \cdot 5^{4}
Sign: 11
Analytic conductor: 1859.261859.26
Root analytic conductor: 6.566526.56652
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 29160000, ( :1/2,1/2), 1)(4,\ 29160000,\ (\ :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
5 1 1
good7C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11D4D_{4} 1+4T+20T2+4pT3+p2T4 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−2T+3T2−2pT3+p2T4 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
19D4D_{4} 1+2T+15T2+2pT3+p2T4 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−4T+44T2−4pT3+p2T4 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+4T+56T2+4pT3+p2T4 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4}
31C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
37D4D_{4} 1+10T+75T2+10pT3+p2T4 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+32T2+4pT3+p2T4 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4}
43C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
47C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
53D4D_{4} 1+4T+104T2+4pT3+p2T4 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
61D4D_{4} 1+2T+27T2+2pT3+p2T4 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−10T+63T2−10pT3+p2T4 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+28T+332T2+28pT3+p2T4 1 + 28 T + 332 T^{2} + 28 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+2T+123T2+2pT3+p2T4 1 + 2 T + 123 T^{2} + 2 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T+135T2−2pT3+p2T4 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−20T+260T2−20pT3+p2T4 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4}
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97D4D_{4} 1+22T+3pT2+22pT3+p2T4 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4}
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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

−7.963459057594579122403119533656, −7.78829570250052895617560346284, −7.17945594294472859161581297922, −6.90653048493958958531445178052, −6.60335726409188197834078815252, −6.26426364323389455850012260092, −5.76295786303059734440524597597, −5.58229840793489948713227592113, −4.92810387794930377824471525324, −4.92577956599006983336175910839, −4.13426275597960821572306435630, −4.06068231896071888977684577272, −3.23756246352056233385789328412, −3.18094941228393760994396172663, −2.48212152249939269578907577761, −2.46396904581188590258298956592, −1.32513231348801711207744572949, −1.32399054697859197599549013288, 0, 0, 1.32399054697859197599549013288, 1.32513231348801711207744572949, 2.46396904581188590258298956592, 2.48212152249939269578907577761, 3.18094941228393760994396172663, 3.23756246352056233385789328412, 4.06068231896071888977684577272, 4.13426275597960821572306435630, 4.92577956599006983336175910839, 4.92810387794930377824471525324, 5.58229840793489948713227592113, 5.76295786303059734440524597597, 6.26426364323389455850012260092, 6.60335726409188197834078815252, 6.90653048493958958531445178052, 7.17945594294472859161581297922, 7.78829570250052895617560346284, 7.963459057594579122403119533656

Graph of the ZZ-function along the critical line