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Properties

Label 16-960e8-1.1-c2e8-0-0
Degree 1616
Conductor 7.214×10237.214\times 10^{23}
Sign 11
Analytic cond. 2.19204×10112.19204\times 10^{11}
Root an. cond. 5.114495.11449
Motivic weight 22
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  + 24·7-s − 12·9-s − 192·23-s + 32·25-s + 144·41-s − 192·47-s + 16·49-s − 288·63-s + 90·81-s − 144·89-s + 72·103-s − 160·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.60e3·161-s + 163-s + 167-s − 992·169-s + 173-s + 768·175-s + 179-s + 181-s + ⋯
L(s)  = 1  + 24/7·7-s − 4/3·9-s − 8.34·23-s + 1.27·25-s + 3.51·41-s − 4.08·47-s + 0.326·49-s − 4.57·63-s + 10/9·81-s − 1.61·89-s + 0.699·103-s − 1.32·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 28.6·161-s + 0.00613·163-s + 0.00598·167-s − 5.86·169-s + 0.00578·173-s + 4.38·175-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

Λ(s)=((248⋅38⋅58)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((248⋅38⋅58)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 248⋅38⋅582^{48} \cdot 3^{8} \cdot 5^{8}
Sign: 11
Analytic conductor: 2.19204×10112.19204\times 10^{11}
Root analytic conductor: 5.114495.11449
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 248⋅38⋅58, ( :[1]8), 1)(16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.375811073×10−61.375811073\times10^{-6}
L(12)L(\frac12) ≈\approx 1.375811073×10−61.375811073\times10^{-6}
L(2)L(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 1
3 (1+pT2)4 ( 1 + p T^{2} )^{4}
5 1−32T2+6p3T4−32p4T6+p8T8 1 - 32 T^{2} + 6 p^{3} T^{4} - 32 p^{4} T^{6} + p^{8} T^{8}
good7 (1−6T+86T2−6p2T3+p4T4)4 ( 1 - 6 T + 86 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{4}
11 (1+80T2+11982T4+80p4T6+p8T8)2 ( 1 + 80 T^{2} + 11982 T^{4} + 80 p^{4} T^{6} + p^{8} T^{8} )^{2}
13 (1+496T2+111822T4+496p4T6+p8T8)2 ( 1 + 496 T^{2} + 111822 T^{4} + 496 p^{4} T^{6} + p^{8} T^{8} )^{2}
17 (1−328T2+23838T4−328p4T6+p8T8)2 ( 1 - 328 T^{2} + 23838 T^{4} - 328 p^{4} T^{6} + p^{8} T^{8} )^{2}
19 (1+290T2+p4T4)4 ( 1 + 290 T^{2} + p^{4} T^{4} )^{4}
23 (1+24T+p2T2)8 ( 1 + 24 T + p^{2} T^{2} )^{8}
29 (1−2864T2+3458382T4−2864p4T6+p8T8)2 ( 1 - 2864 T^{2} + 3458382 T^{4} - 2864 p^{4} T^{6} + p^{8} T^{8} )^{2}
31 (1−2300T2+3121158T4−2300p4T6+p8T8)2 ( 1 - 2300 T^{2} + 3121158 T^{4} - 2300 p^{4} T^{6} + p^{8} T^{8} )^{2}
37 (1+4288T2+8283822T4+4288p4T6+p8T8)2 ( 1 + 4288 T^{2} + 8283822 T^{4} + 4288 p^{4} T^{6} + p^{8} T^{8} )^{2}
41 (1−36T+662T2−36p2T3+p4T4)4 ( 1 - 36 T + 662 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{4}
43 (1−6676T2+17870982T4−6676p4T6+p8T8)2 ( 1 - 6676 T^{2} + 17870982 T^{4} - 6676 p^{4} T^{6} + p^{8} T^{8} )^{2}
47 (1+48T+1970T2+48p2T3+p4T4)4 ( 1 + 48 T + 1970 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4}
53 (1+10384T2+42646350T4+10384p4T6+p8T8)2 ( 1 + 10384 T^{2} + 42646350 T^{4} + 10384 p^{4} T^{6} + p^{8} T^{8} )^{2}
59 (1+8864T2+40876782T4+8864p4T6+p8T8)2 ( 1 + 8864 T^{2} + 40876782 T^{4} + 8864 p^{4} T^{6} + p^{8} T^{8} )^{2}
61 (1+4124T2+16267110T4+4124p4T6+p8T8)2 ( 1 + 4124 T^{2} + 16267110 T^{4} + 4124 p^{4} T^{6} + p^{8} T^{8} )^{2}
67 (1−8020T2+31887942T4−8020p4T6+p8T8)2 ( 1 - 8020 T^{2} + 31887942 T^{4} - 8020 p^{4} T^{6} + p^{8} T^{8} )^{2}
71 (1−7058T2+p4T4)4 ( 1 - 7058 T^{2} + p^{4} T^{4} )^{4}
73 (1−2212T2+45633414T4−2212p4T6+p8T8)2 ( 1 - 2212 T^{2} + 45633414 T^{4} - 2212 p^{4} T^{6} + p^{8} T^{8} )^{2}
79 (1−2108T2−46835706T4−2108p4T6+p8T8)2 ( 1 - 2108 T^{2} - 46835706 T^{4} - 2108 p^{4} T^{6} + p^{8} T^{8} )^{2}
83 (1−25876T2+261715782T4−25876p4T6+p8T8)2 ( 1 - 25876 T^{2} + 261715782 T^{4} - 25876 p^{4} T^{6} + p^{8} T^{8} )^{2}
89 (1+36T+4070T2+36p2T3+p4T4)4 ( 1 + 36 T + 4070 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{4}
97 (1−33364T2+455045286T4−33364p4T6+p8T8)2 ( 1 - 33364 T^{2} + 455045286 T^{4} - 33364 p^{4} T^{6} + p^{8} 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

−4.02367968074451867109428157081, −3.95162827771210956064649499664, −3.82865652880731499502138909521, −3.78891033407257303656407164774, −3.72905091758465551486624786296, −3.56827416586262076375825351313, −3.18648588538343523863501252055, −3.01931943811137150042695005543, −2.93854790363214665690617301907, −2.88553649439195762688543387694, −2.66721798453498855840746549465, −2.41444603893640403015487627129, −2.30983469578774758710312866286, −2.15165743163643032651431336160, −2.08900617646631139884541928224, −1.84177829736740290860433733032, −1.70446541050821082343227499935, −1.67422211712869150076717936323, −1.46902788912707761144656256385, −1.32350056570763926160272140477, −1.23388916234391916951711098978, −0.846719092367898163319440922903, −0.48920124935623255096462357224, −0.05746125831491191508478487654, −0.00055848226386236437406287666, 0.00055848226386236437406287666, 0.05746125831491191508478487654, 0.48920124935623255096462357224, 0.846719092367898163319440922903, 1.23388916234391916951711098978, 1.32350056570763926160272140477, 1.46902788912707761144656256385, 1.67422211712869150076717936323, 1.70446541050821082343227499935, 1.84177829736740290860433733032, 2.08900617646631139884541928224, 2.15165743163643032651431336160, 2.30983469578774758710312866286, 2.41444603893640403015487627129, 2.66721798453498855840746549465, 2.88553649439195762688543387694, 2.93854790363214665690617301907, 3.01931943811137150042695005543, 3.18648588538343523863501252055, 3.56827416586262076375825351313, 3.72905091758465551486624786296, 3.78891033407257303656407164774, 3.82865652880731499502138909521, 3.95162827771210956064649499664, 4.02367968074451867109428157081

Graph of the ZZ-function along the critical line

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