Properties

Label 4-2100e2-1.1-c3e2-0-9
Degree 44
Conductor 44100004410000
Sign 11
Analytic cond. 15352.215352.2
Root an. cond. 11.131211.1312
Motivic weight 33
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  − 9·9-s − 72·11-s + 56·19-s + 372·29-s + 352·31-s − 60·41-s − 49·49-s + 1.12e3·59-s − 644·61-s − 96·71-s + 992·79-s + 81·81-s − 2.62e3·89-s + 648·99-s + 2.07e3·101-s + 1.07e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.23e3·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.97·11-s + 0.676·19-s + 2.38·29-s + 2.03·31-s − 0.228·41-s − 1/7·49-s + 2.48·59-s − 1.35·61-s − 0.160·71-s + 1.41·79-s + 1/9·81-s − 3.12·89-s + 0.657·99-s + 2.04·101-s + 0.945·109-s + 0.921·121-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 + 1.47·169-s + ⋯

Functional equation

Λ(s)=(4410000s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(4410000s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 44100004410000    =    243254722^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 15352.215352.2
Root analytic conductor: 11.131211.1312
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4410000, ( :3/2,3/2), 1)(4,\ 4410000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 3.2393487763.239348776
L(12)L(\frac12) \approx 3.2393487763.239348776
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
3C2C_2 1+p2T2 1 + p^{2} T^{2}
5 1 1
7C2C_2 1+p2T2 1 + p^{2} T^{2}
good11C2C_2 (1+36T+p3T2)2 ( 1 + 36 T + p^{3} T^{2} )^{2}
13C22C_2^2 13238T2+p6T4 1 - 3238 T^{2} + p^{6} T^{4}
17C22C_2^2 19790T2+p6T4 1 - 9790 T^{2} + p^{6} T^{4}
19C2C_2 (128T+p3T2)2 ( 1 - 28 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+12530T2+p6T4 1 + 12530 T^{2} + p^{6} T^{4}
29C2C_2 (1186T+p3T2)2 ( 1 - 186 T + p^{3} T^{2} )^{2}
31C2C_2 (1176T+p3T2)2 ( 1 - 176 T + p^{3} T^{2} )^{2}
37C22C_2^2 1+73418T2+p6T4 1 + 73418 T^{2} + p^{6} T^{4}
41C2C_2 (1+30T+p3T2)2 ( 1 + 30 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+10730T2+p6T4 1 + 10730 T^{2} + p^{6} T^{4}
47C22C_2^2 121022T2+p6T4 1 - 21022 T^{2} + p^{6} T^{4}
53C22C_2^2 1204118T2+p6T4 1 - 204118 T^{2} + p^{6} T^{4}
59C2C_2 (1564T+p3T2)2 ( 1 - 564 T + p^{3} T^{2} )^{2}
61C2C_2 (1+322T+p3T2)2 ( 1 + 322 T + p^{3} T^{2} )^{2}
67C22C_2^2 188870T2+p6T4 1 - 88870 T^{2} + p^{6} T^{4}
71C2C_2 (1+48T+p3T2)2 ( 1 + 48 T + p^{3} T^{2} )^{2}
73C22C_2^2 1+384050T2+p6T4 1 + 384050 T^{2} + p^{6} T^{4}
79C2C_2 (1496T+p3T2)2 ( 1 - 496 T + p^{3} T^{2} )^{2}
83C22C_2^2 1924550T2+p6T4 1 - 924550 T^{2} + p^{6} T^{4}
89C2C_2 (1+1314T+p3T2)2 ( 1 + 1314 T + p^{3} T^{2} )^{2}
97C22C_2^2 1+242498T2+p6T4 1 + 242498 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

−8.734929431542620152009032501132, −8.386176837901720016061457975226, −8.272576904745783059630702332566, −7.935202350023116594696980665213, −7.33748571211491957759774723135, −7.13587130067315372046697426745, −6.47326311175567967049811749956, −6.31150237624100314912831340859, −5.64611996167613717596548255451, −5.42459525683894264807727473101, −4.78683324838058955662225251178, −4.74639242893538630970878805097, −4.16513354780198424437595825245, −3.43150032102024509899743071341, −2.90715903213987040638554415528, −2.74988301121841387995468060550, −2.28627856871080200591304076020, −1.52297557768678847880597076959, −0.70648461417939153523242630146, −0.53662744540970770279638032834, 0.53662744540970770279638032834, 0.70648461417939153523242630146, 1.52297557768678847880597076959, 2.28627856871080200591304076020, 2.74988301121841387995468060550, 2.90715903213987040638554415528, 3.43150032102024509899743071341, 4.16513354780198424437595825245, 4.74639242893538630970878805097, 4.78683324838058955662225251178, 5.42459525683894264807727473101, 5.64611996167613717596548255451, 6.31150237624100314912831340859, 6.47326311175567967049811749956, 7.13587130067315372046697426745, 7.33748571211491957759774723135, 7.935202350023116594696980665213, 8.272576904745783059630702332566, 8.386176837901720016061457975226, 8.734929431542620152009032501132

Graph of the ZZ-function along the critical line