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Properties

Label 4-1100e2-1.1-c1e2-0-3
Degree 44
Conductor 12100001210000
Sign 11
Analytic cond. 77.150677.1506
Root an. cond. 2.963702.96370
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  + 2·9-s − 2·11-s + 8·19-s + 12·29-s + 16·31-s + 12·41-s − 2·49-s + 24·59-s + 4·61-s − 24·71-s − 16·79-s − 5·81-s − 12·89-s − 4·99-s − 36·101-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  + 2/3·9-s − 0.603·11-s + 1.83·19-s + 2.22·29-s + 2.87·31-s + 1.87·41-s − 2/7·49-s + 3.12·59-s + 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s − 1.27·89-s − 0.402·99-s − 3.58·101-s + 1.91·109-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯

Functional equation

Λ(s)=(1210000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1210000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12100001210000    =    24⋅54⋅1122^{4} \cdot 5^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 77.150677.1506
Root analytic conductor: 2.963702.96370
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1210000, ( :1/2,1/2), 1)(4,\ 1210000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7159573802.715957380
L(12)L(\frac12) ≈\approx 2.7159573802.715957380
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
5 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + 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

−9.904619601947231462797622914774, −9.715580398034927405479650009820, −9.554126878613921355380491692879, −8.514919887081907299396917559478, −8.450224077549487878662739886029, −8.204583419709775097730619486966, −7.41231500998221510336887564749, −7.22756490142002398012909377944, −6.82407253323772120423210213616, −6.29833131786111481222644385444, −5.66435842287262031346413980528, −5.53452081378025250626141060079, −4.68887016627501765658936316281, −4.48517842398633277794619898587, −4.08321711411749834182680975996, −3.03417605940130255093621577693, −2.94430395547598527273179454973, −2.35617466104163218299710698300, −1.21163412986945572149062512868, −0.891608224610582192073426463959, 0.891608224610582192073426463959, 1.21163412986945572149062512868, 2.35617466104163218299710698300, 2.94430395547598527273179454973, 3.03417605940130255093621577693, 4.08321711411749834182680975996, 4.48517842398633277794619898587, 4.68887016627501765658936316281, 5.53452081378025250626141060079, 5.66435842287262031346413980528, 6.29833131786111481222644385444, 6.82407253323772120423210213616, 7.22756490142002398012909377944, 7.41231500998221510336887564749, 8.204583419709775097730619486966, 8.450224077549487878662739886029, 8.514919887081907299396917559478, 9.554126878613921355380491692879, 9.715580398034927405479650009820, 9.904619601947231462797622914774

Graph of the ZZ-function along the critical line