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Properties

Label 16-250e8-1.1-c1e8-0-1
Degree 1616
Conductor 1.526×10191.526\times 10^{19}
Sign 11
Analytic cond. 252.195252.195
Root an. cond. 1.412891.41289
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  + 2·2-s − 2·3-s + 4-s − 4·6-s + 16·7-s + 7·9-s − 4·11-s − 2·12-s − 2·13-s + 32·14-s − 4·17-s + 14·18-s − 10·19-s − 32·21-s − 8·22-s − 12·23-s − 4·26-s − 6·27-s + 16·28-s − 10·29-s + 6·31-s − 2·32-s + 8·33-s − 8·34-s + 7·36-s − 4·37-s − 20·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s + 6.04·7-s + 7/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 8.55·14-s − 0.970·17-s + 3.29·18-s − 2.29·19-s − 6.98·21-s − 1.70·22-s − 2.50·23-s − 0.784·26-s − 1.15·27-s + 3.02·28-s − 1.85·29-s + 1.07·31-s − 0.353·32-s + 1.39·33-s − 1.37·34-s + 7/6·36-s − 0.657·37-s − 3.24·38-s + ⋯

Functional equation

Λ(s)=((28⋅524)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅524)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅5242^{8} \cdot 5^{24}
Sign: 11
Analytic conductor: 252.195252.195
Root analytic conductor: 1.412891.41289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅524, ( :[1/2]8), 1)(16,\ 2^{8} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.8316949513.831694951
L(12)L(\frac12) ≈\approx 3.8316949513.831694951
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}
ppFp(T)F_p(T)
bad2 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}
5 1 1
good3 (1−2T+2T2−2pT3+p2T4)2(1+2pT+13T2+10T3+T4+10pT5+13p2T6+2p4T7+p4T8) ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} )
7 (1−8T+6pT2−160T3+471T4−160pT5+6p3T6−8p3T7+p4T8)2 ( 1 - 8 T + 6 p T^{2} - 160 T^{3} + 471 T^{4} - 160 p T^{5} + 6 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
11 1+4T−25T2−80T3+260T4+752T5+123pT6−210pT7−44765T8−210p2T9+123p3T10+752p3T11+260p4T12−80p5T13−25p6T14+4p7T15+p8T16 1 + 4 T - 25 T^{2} - 80 T^{3} + 260 T^{4} + 752 T^{5} + 123 p T^{6} - 210 p T^{7} - 44765 T^{8} - 210 p^{2} T^{9} + 123 p^{3} T^{10} + 752 p^{3} T^{11} + 260 p^{4} T^{12} - 80 p^{5} T^{13} - 25 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
13 1+2T+7T2−34T3+16T4−68T5+985T6−2342T7−9421T8−2342pT9+985p2T10−68p3T11+16p4T12−34p5T13+7p6T14+2p7T15+p8T16 1 + 2 T + 7 T^{2} - 34 T^{3} + 16 T^{4} - 68 T^{5} + 985 T^{6} - 2342 T^{7} - 9421 T^{8} - 2342 p T^{9} + 985 p^{2} T^{10} - 68 p^{3} T^{11} + 16 p^{4} T^{12} - 34 p^{5} T^{13} + 7 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
17 1+4T−pT2−132T3+111T4+364T5−6555T6+8564T7+249544T8+8564pT9−6555p2T10+364p3T11+111p4T12−132p5T13−p7T14+4p7T15+p8T16 1 + 4 T - p T^{2} - 132 T^{3} + 111 T^{4} + 364 T^{5} - 6555 T^{6} + 8564 T^{7} + 249544 T^{8} + 8564 p T^{9} - 6555 p^{2} T^{10} + 364 p^{3} T^{11} + 111 p^{4} T^{12} - 132 p^{5} T^{13} - p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
19 1+10T+22T2+30T3+1023T4+4210T5−4736T6+17200T7+424105T8+17200pT9−4736p2T10+4210p3T11+1023p4T12+30p5T13+22p6T14+10p7T15+p8T16 1 + 10 T + 22 T^{2} + 30 T^{3} + 1023 T^{4} + 4210 T^{5} - 4736 T^{6} + 17200 T^{7} + 424105 T^{8} + 17200 p T^{9} - 4736 p^{2} T^{10} + 4210 p^{3} T^{11} + 1023 p^{4} T^{12} + 30 p^{5} T^{13} + 22 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
23 1+12T+17T2−244T3−304T4+6472T5+29135T6+23748T7−173281T8+23748pT9+29135p2T10+6472p3T11−304p4T12−244p5T13+17p6T14+12p7T15+p8T16 1 + 12 T + 17 T^{2} - 244 T^{3} - 304 T^{4} + 6472 T^{5} + 29135 T^{6} + 23748 T^{7} - 173281 T^{8} + 23748 p T^{9} + 29135 p^{2} T^{10} + 6472 p^{3} T^{11} - 304 p^{4} T^{12} - 244 p^{5} T^{13} + 17 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
29 1+10T+47T2+470T3+4888T4+28460T5+139689T6+999650T7+6593475T8+999650pT9+139689p2T10+28460p3T11+4888p4T12+470p5T13+47p6T14+10p7T15+p8T16 1 + 10 T + 47 T^{2} + 470 T^{3} + 4888 T^{4} + 28460 T^{5} + 139689 T^{6} + 999650 T^{7} + 6593475 T^{8} + 999650 p T^{9} + 139689 p^{2} T^{10} + 28460 p^{3} T^{11} + 4888 p^{4} T^{12} + 470 p^{5} T^{13} + 47 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
31 1−6T+5T2−110T3+1840T4−4048T5−9417T6−113520T7+1708895T8−113520pT9−9417p2T10−4048p3T11+1840p4T12−110p5T13+5p6T14−6p7T15+p8T16 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 113520 p T^{9} - 9417 p^{2} T^{10} - 4048 p^{3} T^{11} + 1840 p^{4} T^{12} - 110 p^{5} T^{13} + 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
37 1+4T−7T2−6pT3+371T4−7606T5−27105T6−1656T7+3727704T8−1656pT9−27105p2T10−7606p3T11+371p4T12−6p6T13−7p6T14+4p7T15+p8T16 1 + 4 T - 7 T^{2} - 6 p T^{3} + 371 T^{4} - 7606 T^{5} - 27105 T^{6} - 1656 T^{7} + 3727704 T^{8} - 1656 p T^{9} - 27105 p^{2} T^{10} - 7606 p^{3} T^{11} + 371 p^{4} T^{12} - 6 p^{6} T^{13} - 7 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
41 1+14T+45T2+130T3+2680T4+7652T5−46977T6−402900T7−2944785T8−402900pT9−46977p2T10+7652p3T11+2680p4T12+130p5T13+45p6T14+14p7T15+p8T16 1 + 14 T + 45 T^{2} + 130 T^{3} + 2680 T^{4} + 7652 T^{5} - 46977 T^{6} - 402900 T^{7} - 2944785 T^{8} - 402900 p T^{9} - 46977 p^{2} T^{10} + 7652 p^{3} T^{11} + 2680 p^{4} T^{12} + 130 p^{5} T^{13} + 45 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
43 (1−14T+158T2−1080T3+7971T4−1080pT5+158p2T6−14p3T7+p4T8)2 ( 1 - 14 T + 158 T^{2} - 1080 T^{3} + 7971 T^{4} - 1080 p T^{5} + 158 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 1+24T+313T2+2808T3+18636T4+1392pT5−279085T6−6389496T7−55226161T8−6389496pT9−279085p2T10+1392p4T11+18636p4T12+2808p5T13+313p6T14+24p7T15+p8T16 1 + 24 T + 313 T^{2} + 2808 T^{3} + 18636 T^{4} + 1392 p T^{5} - 279085 T^{6} - 6389496 T^{7} - 55226161 T^{8} - 6389496 p T^{9} - 279085 p^{2} T^{10} + 1392 p^{4} T^{11} + 18636 p^{4} T^{12} + 2808 p^{5} T^{13} + 313 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16}
53 1−8T−58T2+116T3+127pT4−21448T5−130860T6−424352T7+13016709T8−424352pT9−130860p2T10−21448p3T11+127p5T12+116p5T13−58p6T14−8p7T15+p8T16 1 - 8 T - 58 T^{2} + 116 T^{3} + 127 p T^{4} - 21448 T^{5} - 130860 T^{6} - 424352 T^{7} + 13016709 T^{8} - 424352 p T^{9} - 130860 p^{2} T^{10} - 21448 p^{3} T^{11} + 127 p^{5} T^{12} + 116 p^{5} T^{13} - 58 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
59 1−2pT2+3563T4+324004T6−38801675T8+324004p2T10+3563p4T12−2p7T14+p8T16 1 - 2 p T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 324004 p^{2} T^{10} + 3563 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16}
61 1+14T+30T2−30T3+5215T4+25262T5−186192T6−1508020T7−4396815T8−1508020pT9−186192p2T10+25262p3T11+5215p4T12−30p5T13+30p6T14+14p7T15+p8T16 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 1508020 p T^{9} - 186192 p^{2} T^{10} + 25262 p^{3} T^{11} + 5215 p^{4} T^{12} - 30 p^{5} T^{13} + 30 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
67 1+4T−137T2−192T3+10676T4+15424T5−242695T6+235354T7+7588819T8+235354pT9−242695p2T10+15424p3T11+10676p4T12−192p5T13−137p6T14+4p7T15+p8T16 1 + 4 T - 137 T^{2} - 192 T^{3} + 10676 T^{4} + 15424 T^{5} - 242695 T^{6} + 235354 T^{7} + 7588819 T^{8} + 235354 p T^{9} - 242695 p^{2} T^{10} + 15424 p^{3} T^{11} + 10676 p^{4} T^{12} - 192 p^{5} T^{13} - 137 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
71 1+34T+425T2+2570T3+16960T4+219512T5+2212323T6+19311440T7+171308535T8+19311440pT9+2212323p2T10+219512p3T11+16960p4T12+2570p5T13+425p6T14+34p7T15+p8T16 1 + 34 T + 425 T^{2} + 2570 T^{3} + 16960 T^{4} + 219512 T^{5} + 2212323 T^{6} + 19311440 T^{7} + 171308535 T^{8} + 19311440 p T^{9} + 2212323 p^{2} T^{10} + 219512 p^{3} T^{11} + 16960 p^{4} T^{12} + 2570 p^{5} T^{13} + 425 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16}
73 1+12T−38T2−1764T3−9549T4+121692T5+1406420T6−2622672T7−104550091T8−2622672pT9+1406420p2T10+121692p3T11−9549p4T12−1764p5T13−38p6T14+12p7T15+p8T16 1 + 12 T - 38 T^{2} - 1764 T^{3} - 9549 T^{4} + 121692 T^{5} + 1406420 T^{6} - 2622672 T^{7} - 104550091 T^{8} - 2622672 p T^{9} + 1406420 p^{2} T^{10} + 121692 p^{3} T^{11} - 9549 p^{4} T^{12} - 1764 p^{5} T^{13} - 38 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
79 1−138T2−1500T3+4403T4+205200T5+1430844T6−7779600T7−206126795T8−7779600pT9+1430844p2T10+205200p3T11+4403p4T12−1500p5T13−138p6T14+p8T16 1 - 138 T^{2} - 1500 T^{3} + 4403 T^{4} + 205200 T^{5} + 1430844 T^{6} - 7779600 T^{7} - 206126795 T^{8} - 7779600 p T^{9} + 1430844 p^{2} T^{10} + 205200 p^{3} T^{11} + 4403 p^{4} T^{12} - 1500 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16}
83 1+32T+427T2+2916T3+11596T4+29072T5−1136915T6−33374062T7−418943541T8−33374062pT9−1136915p2T10+29072p3T11+11596p4T12+2916p5T13+427p6T14+32p7T15+p8T16 1 + 32 T + 427 T^{2} + 2916 T^{3} + 11596 T^{4} + 29072 T^{5} - 1136915 T^{6} - 33374062 T^{7} - 418943541 T^{8} - 33374062 p T^{9} - 1136915 p^{2} T^{10} + 29072 p^{3} T^{11} + 11596 p^{4} T^{12} + 2916 p^{5} T^{13} + 427 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16}
89 (1+10T+71T2+1290T3+19001T4+1290pT5+71p2T6+10p3T7+p4T8)2 ( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 1290 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 1+4T−62T2−1692T3+3191T4−2396T5+172940T6−2472896T7+139003669T8−2472896pT9+172940p2T10−2396p3T11+3191p4T12−1692p5T13−62p6T14+4p7T15+p8T16 1 + 4 T - 62 T^{2} - 1692 T^{3} + 3191 T^{4} - 2396 T^{5} + 172940 T^{6} - 2472896 T^{7} + 139003669 T^{8} - 2472896 p T^{9} + 172940 p^{2} T^{10} - 2396 p^{3} T^{11} + 3191 p^{4} T^{12} - 1692 p^{5} T^{13} - 62 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.49093703479943886321986074602, −5.43372771642662983347066278302, −4.98571559554719292923555122886, −4.90462421876633223221319847536, −4.68597308322830525227790415343, −4.59263634450102664309031021697, −4.52686115532177129638187565949, −4.49992812249628717753891832154, −4.36400332421360385316671949810, −4.33737840807449090638038776995, −4.02254084289118645323962448667, −4.00880648930274704342075423455, −4.00010403702497186433442521473, −3.36840450275829870156031297583, −3.13440360938131157882986120569, −2.80198363581217135867929795994, −2.73217128410761034916478451304, −2.64094438855262464013720547792, −2.00150833797728497691990039748, −1.83534057796275142764082466626, −1.66134306922778889926478198438, −1.66041690072960894843552685042, −1.64428550327775767555676573910, −1.56058883963570202201840204955, −0.41756539097360934869493111520, 0.41756539097360934869493111520, 1.56058883963570202201840204955, 1.64428550327775767555676573910, 1.66041690072960894843552685042, 1.66134306922778889926478198438, 1.83534057796275142764082466626, 2.00150833797728497691990039748, 2.64094438855262464013720547792, 2.73217128410761034916478451304, 2.80198363581217135867929795994, 3.13440360938131157882986120569, 3.36840450275829870156031297583, 4.00010403702497186433442521473, 4.00880648930274704342075423455, 4.02254084289118645323962448667, 4.33737840807449090638038776995, 4.36400332421360385316671949810, 4.49992812249628717753891832154, 4.52686115532177129638187565949, 4.59263634450102664309031021697, 4.68597308322830525227790415343, 4.90462421876633223221319847536, 4.98571559554719292923555122886, 5.43372771642662983347066278302, 5.49093703479943886321986074602

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.