Properties

Label 4-882e2-1.1-c5e2-0-14
Degree 44
Conductor 777924777924
Sign 11
Analytic cond. 20010.520010.5
Root an. cond. 11.893611.8936
Motivic weight 55
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  − 8·2-s + 48·4-s + 18·5-s − 256·8-s − 144·10-s − 2·11-s − 288·13-s + 1.28e3·16-s + 1.53e3·17-s − 1.18e3·19-s + 864·20-s + 16·22-s − 3.39e3·23-s − 1.30e3·25-s + 2.30e3·26-s + 3.97e3·29-s − 7.59e3·31-s − 6.14e3·32-s − 1.22e4·34-s + 2.68e3·37-s + 9.50e3·38-s − 4.60e3·40-s + 3.66e4·41-s − 2.30e4·43-s − 96·44-s + 2.71e4·46-s + 864·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.321·5-s − 1.41·8-s − 0.455·10-s − 0.00498·11-s − 0.472·13-s + 5/4·16-s + 1.28·17-s − 0.754·19-s + 0.482·20-s + 0.00704·22-s − 1.33·23-s − 0.416·25-s + 0.668·26-s + 0.877·29-s − 1.41·31-s − 1.06·32-s − 1.81·34-s + 0.322·37-s + 1.06·38-s − 0.455·40-s + 3.40·41-s − 1.89·43-s − 0.00747·44-s + 1.88·46-s + 0.0570·47-s + ⋯

Functional equation

Λ(s)=(777924s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(777924s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 777924777924    =    2234742^{2} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 20010.520010.5
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 777924, ( :5/2,5/2), 1)(4,\ 777924,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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)
bad2C1C_1 (1+p2T)2 ( 1 + p^{2} T )^{2}
3 1 1
7 1 1
good5D4D_{4} 118T+1626T218p5T3+p10T4 1 - 18 T + 1626 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+2T59002T2+2p5T3+p10T4 1 + 2 T - 59002 T^{2} + 2 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+288T+688042T2+288p5T3+p10T4 1 + 288 T + 688042 T^{2} + 288 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 190pT+2366314T290p6T3+p10T4 1 - 90 p T + 2366314 T^{2} - 90 p^{6} T^{3} + p^{10} T^{4}
19D4D_{4} 1+1188T+4834534T2+1188p5T3+p10T4 1 + 1188 T + 4834534 T^{2} + 1188 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+3390T+6218086T2+3390p5T3+p10T4 1 + 3390 T + 6218086 T^{2} + 3390 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 13976T+31254662T23976p5T3+p10T4 1 - 3976 T + 31254662 T^{2} - 3976 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+7596T+61727326T2+7596p5T3+p10T4 1 + 7596 T + 61727326 T^{2} + 7596 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 12688T+126774470T22688p5T3+p10T4 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 136630T+559995322T236630p5T3+p10T4 1 - 36630 T + 559995322 T^{2} - 36630 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+23032T+329072262T2+23032p5T3+p10T4 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1864T+46643358T2864p5T3+p10T4 1 - 864 T + 46643358 T^{2} - 864 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 132920T+983844566T232920p5T3+p10T4 1 - 32920 T + 983844566 T^{2} - 32920 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+26712T+697343334T2+26712p5T3+p10T4 1 + 26712 T + 697343334 T^{2} + 26712 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+20412T+1230091258T2+20412p5T3+p10T4 1 + 20412 T + 1230091258 T^{2} + 20412 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+36172T+33148290pT2+36172p5T3+p10T4 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+73706T+3356433686T2+73706p5T3+p10T4 1 + 73706 T + 3356433686 T^{2} + 73706 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 174772T+3993671602T274772p5T3+p10T4 1 - 74772 T + 3993671602 T^{2} - 74772 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+23116T+6249589662T2+23116p5T3+p10T4 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1147816T+13316083030T2147816p5T3+p10T4 1 - 147816 T + 13316083030 T^{2} - 147816 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1164646T+17878567722T2164646p5T3+p10T4 1 - 164646 T + 17878567722 T^{2} - 164646 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+162036T+20528488258T2+162036p5T3+p10T4 1 + 162036 T + 20528488258 T^{2} + 162036 p^{5} T^{3} + p^{10} 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

−9.110987749020565277903405329118, −8.991401496919358282146114694328, −8.220400334419701025294000155163, −8.002081889761395316410637999759, −7.55430958542966741602841448909, −7.37080987351475880579049186904, −6.57017348092223294919284127802, −6.35577450239813307539667345171, −5.66512933122514901580679591304, −5.63181366803601933671415933532, −4.77707166905828990550874416558, −4.16936035130082708753465584189, −3.65836644967760464957873419020, −3.04556915047309937487484153527, −2.28413930294448249988341959805, −2.22497586143914281034170014326, −1.30001818799649595666258386704, −1.04420409080615409248609403174, 0, 0, 1.04420409080615409248609403174, 1.30001818799649595666258386704, 2.22497586143914281034170014326, 2.28413930294448249988341959805, 3.04556915047309937487484153527, 3.65836644967760464957873419020, 4.16936035130082708753465584189, 4.77707166905828990550874416558, 5.63181366803601933671415933532, 5.66512933122514901580679591304, 6.35577450239813307539667345171, 6.57017348092223294919284127802, 7.37080987351475880579049186904, 7.55430958542966741602841448909, 8.002081889761395316410637999759, 8.220400334419701025294000155163, 8.991401496919358282146114694328, 9.110987749020565277903405329118

Graph of the ZZ-function along the critical line