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Properties

Label 16-87e16-1.1-c1e8-0-2
Degree 1616
Conductor 1.077×10311.077\times 10^{31}
Sign 11
Analytic cond. 1.78042×10141.78042\times 10^{14}
Root an. cond. 7.774237.77423
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  − 5·4-s + 2·13-s + 9·16-s + 20·19-s − 19·25-s + 10·31-s + 18·37-s + 28·43-s − 26·49-s − 10·52-s + 54·61-s − 10·64-s + 38·67-s − 16·73-s − 100·76-s + 14·79-s − 6·97-s + 95·100-s − 14·103-s + 2·109-s − 34·121-s − 50·124-s + 127-s + 131-s + 137-s + 139-s − 90·148-s + ⋯
L(s)  = 1  − 5/2·4-s + 0.554·13-s + 9/4·16-s + 4.58·19-s − 3.79·25-s + 1.79·31-s + 2.95·37-s + 4.26·43-s − 3.71·49-s − 1.38·52-s + 6.91·61-s − 5/4·64-s + 4.64·67-s − 1.87·73-s − 11.4·76-s + 1.57·79-s − 0.609·97-s + 19/2·100-s − 1.37·103-s + 0.191·109-s − 3.09·121-s − 4.49·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.39·148-s + ⋯

Functional equation

Λ(s)=((316⋅2916)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((316⋅2916)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 316⋅29163^{16} \cdot 29^{16}
Sign: 11
Analytic conductor: 1.78042×10141.78042\times 10^{14}
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 316⋅2916, ( :[1/2]8), 1)(16,\ 3^{16} \cdot 29^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 18.5265725318.52657253
L(12)L(\frac12) ≈\approx 18.5265725318.52657253
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)
bad3 1 1
29 1 1
good2 1+5T2+p4T4+45T6+101T8+45p2T10+p8T12+5p6T14+p8T16 1 + 5 T^{2} + p^{4} T^{4} + 45 T^{6} + 101 T^{8} + 45 p^{2} T^{10} + p^{8} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16}
5 1+19T2+191T4+1429T6+8256T8+1429p2T10+191p4T12+19p6T14+p8T16 1 + 19 T^{2} + 191 T^{4} + 1429 T^{6} + 8256 T^{8} + 1429 p^{2} T^{10} + 191 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16}
7 (1+13T2+20T3+89T4+20pT5+13p2T6+p4T8)2 ( 1 + 13 T^{2} + 20 T^{3} + 89 T^{4} + 20 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2}
11 1+34T2+645T4+8726T6+9004pT8+8726p2T10+645p4T12+34p6T14+p8T16 1 + 34 T^{2} + 645 T^{4} + 8726 T^{6} + 9004 p T^{8} + 8726 p^{2} T^{10} + 645 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16}
13 (1−T+38T2−30T3+651T4−30pT5+38p2T6−p3T7+p4T8)2 ( 1 - T + 38 T^{2} - 30 T^{3} + 651 T^{4} - 30 p T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
17 1+52T2+1460T4+30092T6+527254T8+30092p2T10+1460p4T12+52p6T14+p8T16 1 + 52 T^{2} + 1460 T^{4} + 30092 T^{6} + 527254 T^{8} + 30092 p^{2} T^{10} + 1460 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16}
19 (1−5T+43T2−5pT3+p2T4)4 ( 1 - 5 T + 43 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4}
23 1+35T2+1571T4+39685T6+1089016T8+39685p2T10+1571p4T12+35p6T14+p8T16 1 + 35 T^{2} + 1571 T^{4} + 39685 T^{6} + 1089016 T^{8} + 39685 p^{2} T^{10} + 1571 p^{4} T^{12} + 35 p^{6} T^{14} + p^{8} T^{16}
31 (1−5T+114T2−460T3+5151T4−460pT5+114p2T6−5p3T7+p4T8)2 ( 1 - 5 T + 114 T^{2} - 460 T^{3} + 5151 T^{4} - 460 p T^{5} + 114 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
37 (1−9T+109T2−673T3+5324T4−673pT5+109p2T6−9p3T7+p4T8)2 ( 1 - 9 T + 109 T^{2} - 673 T^{3} + 5324 T^{4} - 673 p T^{5} + 109 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2}
41 1+162T2+14413T4+889934T6+41235380T8+889934p2T10+14413p4T12+162p6T14+p8T16 1 + 162 T^{2} + 14413 T^{4} + 889934 T^{6} + 41235380 T^{8} + 889934 p^{2} T^{10} + 14413 p^{4} T^{12} + 162 p^{6} T^{14} + p^{8} T^{16}
43 (1−14T+103T2−960T3+8201T4−960pT5+103p2T6−14p3T7+p4T8)2 ( 1 - 14 T + 103 T^{2} - 960 T^{3} + 8201 T^{4} - 960 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 1+155T2+12331T4+782445T6+41499176T8+782445p2T10+12331p4T12+155p6T14+p8T16 1 + 155 T^{2} + 12331 T^{4} + 782445 T^{6} + 41499176 T^{8} + 782445 p^{2} T^{10} + 12331 p^{4} T^{12} + 155 p^{6} T^{14} + p^{8} T^{16}
53 (1+144T2+10622T4+144p2T6+p4T8)2 ( 1 + 144 T^{2} + 10622 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2}
59 1+211T2+24975T4+2115569T6+138442424T8+2115569p2T10+24975p4T12+211p6T14+p8T16 1 + 211 T^{2} + 24975 T^{4} + 2115569 T^{6} + 138442424 T^{8} + 2115569 p^{2} T^{10} + 24975 p^{4} T^{12} + 211 p^{6} T^{14} + p^{8} T^{16}
61 (1−27T+498T2−5874T3+54255T4−5874pT5+498p2T6−27p3T7+p4T8)2 ( 1 - 27 T + 498 T^{2} - 5874 T^{3} + 54255 T^{4} - 5874 p T^{5} + 498 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2}
67 (1−19T+304T2−2838T3+27239T4−2838pT5+304p2T6−19p3T7+p4T8)2 ( 1 - 19 T + 304 T^{2} - 2838 T^{3} + 27239 T^{4} - 2838 p T^{5} + 304 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2}
71 1+103T2+17543T4+1340981T6+130159480T8+1340981p2T10+17543p4T12+103p6T14+p8T16 1 + 103 T^{2} + 17543 T^{4} + 1340981 T^{6} + 130159480 T^{8} + 1340981 p^{2} T^{10} + 17543 p^{4} T^{12} + 103 p^{6} T^{14} + p^{8} T^{16}
73 (1+8T+131T2+1094T3+14619T4+1094pT5+131p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 131 T^{2} + 1094 T^{3} + 14619 T^{4} + 1094 p T^{5} + 131 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
79 (1−7T+220T2−1272T3+24269T4−1272pT5+220p2T6−7p3T7+p4T8)2 ( 1 - 7 T + 220 T^{2} - 1272 T^{3} + 24269 T^{4} - 1272 p T^{5} + 220 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2}
83 1+138T2+19885T4+1799198T6+178905044T8+1799198p2T10+19885p4T12+138p6T14+p8T16 1 + 138 T^{2} + 19885 T^{4} + 1799198 T^{6} + 178905044 T^{8} + 1799198 p^{2} T^{10} + 19885 p^{4} T^{12} + 138 p^{6} T^{14} + p^{8} T^{16}
89 1+438T2+93973T4+13385366T6+1386133340T8+13385366p2T10+93973p4T12+438p6T14+p8T16 1 + 438 T^{2} + 93973 T^{4} + 13385366 T^{6} + 1386133340 T^{8} + 13385366 p^{2} T^{10} + 93973 p^{4} T^{12} + 438 p^{6} T^{14} + p^{8} T^{16}
97 (1+3T+167T2+5pT3+18096T4+5p2T5+167p2T6+3p3T7+p4T8)2 ( 1 + 3 T + 167 T^{2} + 5 p T^{3} + 18096 T^{4} + 5 p^{2} T^{5} + 167 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
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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

−3.25587017742218836587592774088, −3.01436607424833740306909198472, −2.98456094925968581543986238385, −2.94787983822079500233073119316, −2.78854821328600252291656389107, −2.65278016016051110062197805639, −2.49313691594091675791613929423, −2.43831319009164804275880040193, −2.35267056768649447646175877615, −2.29529959927873990136212201101, −2.23701910420667289000917495422, −1.79545310967620180218745117614, −1.70924272134575210928263780891, −1.68973857248096997756552697184, −1.67368061418384315615505288318, −1.52832090849494420631448839289, −1.32718891362557278936359078297, −1.00693681199066280822759296783, −0.891200533339698870697225726519, −0.797915353672243560754955172632, −0.74042989821156927320148775086, −0.61915879484559102634809750169, −0.54797940048731320516837461728, −0.51927517703419244216967404636, −0.24122970998436824536408134927, 0.24122970998436824536408134927, 0.51927517703419244216967404636, 0.54797940048731320516837461728, 0.61915879484559102634809750169, 0.74042989821156927320148775086, 0.797915353672243560754955172632, 0.891200533339698870697225726519, 1.00693681199066280822759296783, 1.32718891362557278936359078297, 1.52832090849494420631448839289, 1.67368061418384315615505288318, 1.68973857248096997756552697184, 1.70924272134575210928263780891, 1.79545310967620180218745117614, 2.23701910420667289000917495422, 2.29529959927873990136212201101, 2.35267056768649447646175877615, 2.43831319009164804275880040193, 2.49313691594091675791613929423, 2.65278016016051110062197805639, 2.78854821328600252291656389107, 2.94787983822079500233073119316, 2.98456094925968581543986238385, 3.01436607424833740306909198472, 3.25587017742218836587592774088

Graph of the ZZ-function along the critical line

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