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Properties

Label 4-700e2-1.1-c3e2-0-11
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 1705.801705.80
Root an. cond. 6.426616.42661
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  − 14·7-s − 48·9-s + 48·11-s + 28·13-s + 28·17-s − 88·19-s − 28·23-s − 140·29-s + 104·31-s + 364·37-s + 180·41-s + 392·43-s + 56·47-s + 147·49-s + 924·53-s + 568·59-s − 468·61-s + 672·63-s + 224·67-s − 972·71-s + 1.31e3·73-s − 672·77-s + 964·79-s + 1.57e3·81-s + 672·83-s − 1.65e3·89-s − 392·91-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.77·9-s + 1.31·11-s + 0.597·13-s + 0.399·17-s − 1.06·19-s − 0.253·23-s − 0.896·29-s + 0.602·31-s + 1.61·37-s + 0.685·41-s + 1.39·43-s + 0.173·47-s + 3/7·49-s + 2.39·53-s + 1.25·59-s − 0.982·61-s + 1.34·63-s + 0.408·67-s − 1.62·71-s + 2.10·73-s − 0.994·77-s + 1.37·79-s + 2.16·81-s + 0.888·83-s − 1.96·89-s − 0.451·91-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 1705.801705.80
Root analytic conductor: 6.426616.42661
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :3/2,3/2), 1)(4,\ 490000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5624438162.562443816
L(12)L(\frac12) ≈\approx 2.5624438162.562443816
L(52)L(\frac{5}{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
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1+16pT2+p6T4 1 + 16 p T^{2} + p^{6} T^{4}
11D4D_{4} 1−48T+2062T2−48p3T3+p6T4 1 - 48 T + 2062 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−28T+1416T2−28p3T3+p6T4 1 - 28 T + 1416 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−28T+6566T2−28p3T3+p6T4 1 - 28 T + 6566 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+88T+15360T2+88p3T3+p6T4 1 + 88 T + 15360 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+28T+24506T2+28p3T3+p6T4 1 + 28 T + 24506 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+140T+52502T2+140p3T3+p6T4 1 + 140 T + 52502 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−104T+61110T2−104p3T3+p6T4 1 - 104 T + 61110 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−364T+120606T2−364p3T3+p6T4 1 - 364 T + 120606 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−180T+3646T2−180p3T3+p6T4 1 - 180 T + 3646 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−392T+144414T2−392p3T3+p6T4 1 - 392 T + 144414 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−56T+179030T2−56p3T3+p6T4 1 - 56 T + 179030 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−924T+496198T2−924p3T3+p6T4 1 - 924 T + 496198 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−568T+335888T2−568p3T3+p6T4 1 - 568 T + 335888 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+468T+353192T2+468p3T3+p6T4 1 + 468 T + 353192 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−224T+116406T2−224p3T3+p6T4 1 - 224 T + 116406 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+972T+950842T2+972p3T3+p6T4 1 + 972 T + 950842 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−1316T+1153374T2−1316p3T3+p6T4 1 - 1316 T + 1153374 T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−964T+483402T2−964p3T3+p6T4 1 - 964 T + 483402 T^{2} - 964 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−672T+762256T2−672p3T3+p6T4 1 - 672 T + 762256 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1652T+1922870T2+1652p3T3+p6T4 1 + 1652 T + 1922870 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1428T+2332742T2−1428p3T3+p6T4 1 - 1428 T + 2332742 T^{2} - 1428 p^{3} T^{3} + p^{6} 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

−10.14690380272769903734412310835, −9.878476075799528725119323553421, −9.269297340248471613425645332889, −8.888353985389832936604863886299, −8.719077325368560915412978530928, −8.290177435948289459891489228221, −7.46642677453915107889095411878, −7.42950945932524763171549843435, −6.38157856652778788813751105292, −6.33336649154832467263341022022, −5.87985077051909231196385910517, −5.59547321506238545669936287696, −4.71180909429618519957168465504, −4.19631100585705602211265960483, −3.56915392624490267661698196495, −3.37891067844283899104707359869, −2.32244297973380705804394497343, −2.29476537170809633670083383789, −0.906490646171834012620050779679, −0.58064542603283174191661162934, 0.58064542603283174191661162934, 0.906490646171834012620050779679, 2.29476537170809633670083383789, 2.32244297973380705804394497343, 3.37891067844283899104707359869, 3.56915392624490267661698196495, 4.19631100585705602211265960483, 4.71180909429618519957168465504, 5.59547321506238545669936287696, 5.87985077051909231196385910517, 6.33336649154832467263341022022, 6.38157856652778788813751105292, 7.42950945932524763171549843435, 7.46642677453915107889095411878, 8.290177435948289459891489228221, 8.719077325368560915412978530928, 8.888353985389832936604863886299, 9.269297340248471613425645332889, 9.878476075799528725119323553421, 10.14690380272769903734412310835

Graph of the ZZ-function along the critical line