Properties

Label 4-1800e2-1.1-c3e2-0-13
Degree 44
Conductor 32400003240000
Sign 11
Analytic cond. 11279.111279.1
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 144·11-s − 104·19-s − 156·29-s + 240·31-s − 724·41-s + 670·49-s + 1.39e3·59-s + 444·61-s − 192·71-s + 1.26e3·79-s + 1.98e3·89-s − 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.94·11-s − 1.25·19-s − 0.998·29-s + 1.39·31-s − 2.75·41-s + 1.95·49-s + 3.07·59-s + 0.931·61-s − 0.320·71-s + 1.80·79-s + 2.36·89-s − 1.75·101-s − 0.783·109-s + 9.68·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯

Functional equation

Λ(s)=(3240000s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(3240000s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 32400003240000    =    2634542^{6} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 11279.111279.1
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3240000, ( :3/2,3/2), 1)(4,\ 3240000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 1.5043056271.504305627
L(12)L(\frac12) \approx 1.5043056271.504305627
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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
good7C22C_2^2 1670T2+p6T4 1 - 670 T^{2} + p^{6} T^{4}
11C2C_2 (1+72T+p3T2)2 ( 1 + 72 T + p^{3} T^{2} )^{2}
13C22C_2^2 14358T2+p6T4 1 - 4358 T^{2} + p^{6} T^{4}
17C22C_2^2 18382T2+p6T4 1 - 8382 T^{2} + p^{6} T^{4}
19C2C_2 (1+52T+p3T2)2 ( 1 + 52 T + p^{3} T^{2} )^{2}
23C22C_2^2 11230T2+p6T4 1 - 1230 T^{2} + p^{6} T^{4}
29C2C_2 (1+78T+p3T2)2 ( 1 + 78 T + p^{3} T^{2} )^{2}
31C2C_2 (1120T+p3T2)2 ( 1 - 120 T + p^{3} T^{2} )^{2}
37C22C_2^2 178806T2+p6T4 1 - 78806 T^{2} + p^{6} T^{4}
41C2C_2 (1+362T+p3T2)2 ( 1 + 362 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+75242T2+p6T4 1 + 75242 T^{2} + p^{6} T^{4}
47C22C_2^2 1129246T2+p6T4 1 - 129246 T^{2} + p^{6} T^{4}
53C22C_2^2 1+151146T2+p6T4 1 + 151146 T^{2} + p^{6} T^{4}
59C2C_2 (1696T+p3T2)2 ( 1 - 696 T + p^{3} T^{2} )^{2}
61C2C_2 (1222T+p3T2)2 ( 1 - 222 T + p^{3} T^{2} )^{2}
67C22C_2^2 1601510T2+p6T4 1 - 601510 T^{2} + p^{6} T^{4}
71C2C_2 (1+96T+p3T2)2 ( 1 + 96 T + p^{3} T^{2} )^{2}
73C22C_2^2 1746350T2+p6T4 1 - 746350 T^{2} + p^{6} T^{4}
79C2C_2 (18pT+p3T2)2 ( 1 - 8 p T + p^{3} T^{2} )^{2}
83C22C_2^2 1769030T2+p6T4 1 - 769030 T^{2} + p^{6} T^{4}
89C2C_2 (1994T+p3T2)2 ( 1 - 994 T + p^{3} T^{2} )^{2}
97C22C_2^2 1+844610T2+p6T4 1 + 844610 T^{2} + p^{6} T^{4}
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   L(s)=p j=14(1αj,pps)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

−8.769624283951839779224161818360, −8.729506132722121740935377533976, −8.252461804679939124160849011687, −8.021432699691598130693618774215, −7.47366694378677567527593638023, −7.41846790654937294001570853889, −6.61374709425533344248739659561, −6.47945172842430259768838614561, −5.67865959901195474963839717366, −5.38820423823188472610113684890, −5.04527079815134221666517897767, −4.93764322837246195963958473084, −4.04455543653042023952337869989, −3.77153685162631980518425940141, −2.86872127922451614923331089956, −2.77872204378069124863887952772, −2.12657878612317860353125078668, −1.95035263958724987888613994055, −0.60467369287496069402479059828, −0.40935333957269288356934686551, 0.40935333957269288356934686551, 0.60467369287496069402479059828, 1.95035263958724987888613994055, 2.12657878612317860353125078668, 2.77872204378069124863887952772, 2.86872127922451614923331089956, 3.77153685162631980518425940141, 4.04455543653042023952337869989, 4.93764322837246195963958473084, 5.04527079815134221666517897767, 5.38820423823188472610113684890, 5.67865959901195474963839717366, 6.47945172842430259768838614561, 6.61374709425533344248739659561, 7.41846790654937294001570853889, 7.47366694378677567527593638023, 8.021432699691598130693618774215, 8.252461804679939124160849011687, 8.729506132722121740935377533976, 8.769624283951839779224161818360

Graph of the ZZ-function along the critical line