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Properties

Label 16-882e8-1.1-c1e8-0-2
Degree 1616
Conductor 3.662×10233.662\times 10^{23}
Sign 11
Analytic cond. 6.05292×1066.05292\times 10^{6}
Root an. cond. 2.653822.65382
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  − 8·2-s + 36·4-s − 120·8-s − 8·11-s + 330·16-s + 64·22-s − 4·23-s + 16·25-s − 4·29-s − 792·32-s + 32·37-s + 28·43-s − 288·44-s + 32·46-s − 128·50-s − 20·53-s + 32·58-s + 1.71e3·64-s − 72·67-s − 48·71-s − 256·74-s + 16·79-s + 9·81-s − 224·86-s + 960·88-s − 144·92-s + 576·100-s + ⋯
L(s)  = 1  − 5.65·2-s + 18·4-s − 42.4·8-s − 2.41·11-s + 82.5·16-s + 13.6·22-s − 0.834·23-s + 16/5·25-s − 0.742·29-s − 140.·32-s + 5.26·37-s + 4.26·43-s − 43.4·44-s + 4.71·46-s − 18.1·50-s − 2.74·53-s + 4.20·58-s + 214.5·64-s − 8.79·67-s − 5.69·71-s − 29.7·74-s + 1.80·79-s + 81-s − 24.1·86-s + 102.·88-s − 15.0·92-s + 57.5·100-s + ⋯

Functional equation

Λ(s)=((28⋅316⋅716)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅316⋅716)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅316⋅7162^{8} \cdot 3^{16} \cdot 7^{16}
Sign: 11
Analytic conductor: 6.05292×1066.05292\times 10^{6}
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅316⋅716, ( :[1/2]8), 1)(16,\ 2^{8} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.069846463920.06984646392
L(12)L(\frac12) ≈\approx 0.069846463920.06984646392
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)8 ( 1 + T )^{8}
3 1−p2T4+p4T8 1 - p^{2} T^{4} + p^{4} T^{8}
7 1 1
good5 1−16T2+29pT4−976T6+5296T8−976p2T10+29p5T12−16p6T14+p8T16 1 - 16 T^{2} + 29 p T^{4} - 976 T^{6} + 5296 T^{8} - 976 p^{2} T^{10} + 29 p^{5} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16}
11 (1+4T+2T2−32T3−101T4−32pT5+2p2T6+4p3T7+p4T8)2 ( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
13 (1−20T2+231T4−20p2T6+p4T8)2 ( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2}
17 1−16T2−194T4+2048T6+44995T8+2048p2T10−194p4T12−16p6T14+p8T16 1 - 16 T^{2} - 194 T^{4} + 2048 T^{6} + 44995 T^{8} + 2048 p^{2} T^{10} - 194 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16}
19 1−72T2+3169T4−93096T6+2058480T8−93096p2T10+3169p4T12−72p6T14+p8T16 1 - 72 T^{2} + 3169 T^{4} - 93096 T^{6} + 2058480 T^{8} - 93096 p^{2} T^{10} + 3169 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16}
23 (1+2T+5T2−94T3−620T4−94pT5+5p2T6+2p3T7+p4T8)2 ( 1 + 2 T + 5 T^{2} - 94 T^{3} - 620 T^{4} - 94 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1+2T−52T2−4T3+2179T4−4pT5−52p2T6+2p3T7+p4T8)2 ( 1 + 2 T - 52 T^{2} - 4 T^{3} + 2179 T^{4} - 4 p T^{5} - 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
31 (1+8T2+p2T4)4 ( 1 + 8 T^{2} + p^{2} T^{4} )^{4}
37 (1−8T+27T2−8pT3+p2T4)4 ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4}
41 (1−50T2+819T4−50p2T6+p4T8)2 ( 1 - 50 T^{2} + 819 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2}
43 (1−14T+88T2−308T3+1387T4−308pT5+88p2T6−14p3T7+p4T8)2 ( 1 - 14 T + 88 T^{2} - 308 T^{3} + 1387 T^{4} - 308 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 (1+76T2+4662T4+76p2T6+p4T8)2 ( 1 + 76 T^{2} + 4662 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2}
53 (1+10T−28T2+220T3+8275T4+220pT5−28p2T6+10p3T7+p4T8)2 ( 1 + 10 T - 28 T^{2} + 220 T^{3} + 8275 T^{4} + 220 p T^{5} - 28 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
59 (1−8T2−2430T4−8p2T6+p4T8)2 ( 1 - 8 T^{2} - 2430 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2}
61 (1+216T2+19079T4+216p2T6+p4T8)2 ( 1 + 216 T^{2} + 19079 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1+18T+188T2+18pT3+p2T4)4 ( 1 + 18 T + 188 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4}
71 (1+12T+151T2+12pT3+p2T4)4 ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4}
73 1−180T2+14842T4−1242000T6+107225523T8−1242000p2T10+14842p4T12−180p6T14+p8T16 1 - 180 T^{2} + 14842 T^{4} - 1242000 T^{6} + 107225523 T^{8} - 1242000 p^{2} T^{10} + 14842 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16}
79 (1−4T+15T2−4pT3+p2T4)4 ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4}
83 (1−68T2−2265T4−68p2T6+p4T8)2 ( 1 - 68 T^{2} - 2265 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 1+80T2−9314T4−10240T6+133493155T8−10240p2T10−9314p4T12+80p6T14+p8T16 1 + 80 T^{2} - 9314 T^{4} - 10240 T^{6} + 133493155 T^{8} - 10240 p^{2} T^{10} - 9314 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16}
97 1−372T2+85018T4−12851856T6+1457345619T8−12851856p2T10+85018p4T12−372p6T14+p8T16 1 - 372 T^{2} + 85018 T^{4} - 12851856 T^{6} + 1457345619 T^{8} - 12851856 p^{2} T^{10} + 85018 p^{4} T^{12} - 372 p^{6} T^{14} + 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

−4.51612564643574825928855110575, −4.30424711503623088028217870070, −4.09051029625363667770854670515, −3.80657966632598295180111133303, −3.77122631588711922113128629313, −3.51822023304080917422853131159, −3.14502808738321113603691671305, −3.09262777402254045510598126952, −2.97426478016247542695006481187, −2.84803711208535615590004090211, −2.82291491530787779788584744884, −2.72796142768427488678440137426, −2.60306195195237913732946795167, −2.56809068636433414786694054518, −2.25614119793965932513469746377, −2.01268948403321958875719025800, −1.77474860615799703809459296091, −1.74046208870896397572591369528, −1.48740179396304525610555277719, −1.35733940898393730287615998155, −1.12493508923873528824968270482, −0.912002545428551141285855067777, −0.69178018853338750252696877068, −0.45682979462238907134217828964, −0.16408342159455284089235721150, 0.16408342159455284089235721150, 0.45682979462238907134217828964, 0.69178018853338750252696877068, 0.912002545428551141285855067777, 1.12493508923873528824968270482, 1.35733940898393730287615998155, 1.48740179396304525610555277719, 1.74046208870896397572591369528, 1.77474860615799703809459296091, 2.01268948403321958875719025800, 2.25614119793965932513469746377, 2.56809068636433414786694054518, 2.60306195195237913732946795167, 2.72796142768427488678440137426, 2.82291491530787779788584744884, 2.84803711208535615590004090211, 2.97426478016247542695006481187, 3.09262777402254045510598126952, 3.14502808738321113603691671305, 3.51822023304080917422853131159, 3.77122631588711922113128629313, 3.80657966632598295180111133303, 4.09051029625363667770854670515, 4.30424711503623088028217870070, 4.51612564643574825928855110575

Graph of the ZZ-function along the critical line

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