Properties

Label 16-12e24-1.1-c2e8-0-6
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 16·13-s + 48·17-s − 44·25-s + 32·29-s + 96·37-s − 128·41-s + 196·49-s − 168·53-s − 32·61-s − 128·65-s + 24·73-s − 384·85-s + 624·89-s − 136·97-s − 8·101-s − 192·109-s − 640·113-s + 364·121-s + 192·125-s + 127-s + 131-s + 137-s + 139-s − 256·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 8/5·5-s + 1.23·13-s + 2.82·17-s − 1.75·25-s + 1.10·29-s + 2.59·37-s − 3.12·41-s + 4·49-s − 3.16·53-s − 0.524·61-s − 1.96·65-s + 0.328·73-s − 4.51·85-s + 7.01·89-s − 1.40·97-s − 0.0792·101-s − 1.76·109-s − 5.66·113-s + 3.00·121-s + 1.53·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.76·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1758830584\)
\(L(\frac12)\) \(\approx\) \(0.1758830584\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 + 4 T + 46 T^{2} + 328 T^{3} + 1279 T^{4} + 328 p^{2} T^{5} + 46 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
7 \( 1 - 4 p^{2} T^{2} + 17362 T^{4} - 998752 T^{6} + 49146619 T^{8} - 998752 p^{4} T^{10} + 17362 p^{8} T^{12} - 4 p^{14} T^{14} + p^{16} T^{16} \)
11 \( 1 - 364 T^{2} + 100090 T^{4} - 17908816 T^{6} + 2533594627 T^{8} - 17908816 p^{4} T^{10} + 100090 p^{8} T^{12} - 364 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 - 8 T + 388 T^{2} - 3416 T^{3} + 89638 T^{4} - 3416 p^{2} T^{5} + 388 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 24 T + 1060 T^{2} - 18504 T^{3} + 454854 T^{4} - 18504 p^{2} T^{5} + 1060 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1688 T^{2} + 1410076 T^{4} - 41454776 p T^{6} + 326011377286 T^{8} - 41454776 p^{5} T^{10} + 1410076 p^{8} T^{12} - 1688 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 2104 T^{2} + 2402716 T^{4} - 1927306888 T^{6} + 1169609629126 T^{8} - 1927306888 p^{4} T^{10} + 2402716 p^{8} T^{12} - 2104 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 16 T + 2716 T^{2} - 43888 T^{3} + 3151462 T^{4} - 43888 p^{2} T^{5} + 2716 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 4468 T^{2} + 9741058 T^{4} - 14171350144 T^{6} + 15470259465835 T^{8} - 14171350144 p^{4} T^{10} + 9741058 p^{8} T^{12} - 4468 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 48 T + 1996 T^{2} - 55440 T^{3} + 1362630 T^{4} - 55440 p^{2} T^{5} + 1996 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 + 64 T + 4348 T^{2} + 112576 T^{3} + 5748550 T^{4} + 112576 p^{2} T^{5} + 4348 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 7096 T^{2} + 21143260 T^{4} - 33679978888 T^{6} + 47262929427334 T^{8} - 33679978888 p^{4} T^{10} + 21143260 p^{8} T^{12} - 7096 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 10456 T^{2} + 55036252 T^{4} - 191495160040 T^{6} + 487918139630278 T^{8} - 191495160040 p^{4} T^{10} + 55036252 p^{8} T^{12} - 10456 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 84 T + 9286 T^{2} + 546984 T^{3} + 34768983 T^{4} + 546984 p^{2} T^{5} + 9286 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 11656 T^{2} + 91670236 T^{4} - 474544006456 T^{6} + 1929452638722694 T^{8} - 474544006456 p^{4} T^{10} + 91670236 p^{8} T^{12} - 11656 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 16 T + 3748 T^{2} - 35408 T^{3} + 3940582 T^{4} - 35408 p^{2} T^{5} + 3748 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 12088 T^{2} + 77794396 T^{4} - 475176793864 T^{6} + 2559979434564358 T^{8} - 475176793864 p^{4} T^{10} + 77794396 p^{8} T^{12} - 12088 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 24712 T^{2} + 317520028 T^{4} - 2663333105848 T^{6} + 15809843114733766 T^{8} - 2663333105848 p^{4} T^{10} + 317520028 p^{8} T^{12} - 24712 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 12 T + 10690 T^{2} + 373824 T^{3} + 49428075 T^{4} + 373824 p^{2} T^{5} + 10690 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 20488 T^{2} + 292556572 T^{4} - 2710137606712 T^{6} + 19925422059595078 T^{8} - 2710137606712 p^{4} T^{10} + 292556572 p^{8} T^{12} - 20488 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 19180 T^{2} + 272112634 T^{4} - 2832260708560 T^{6} + 21354167069332099 T^{8} - 2832260708560 p^{4} T^{10} + 272112634 p^{8} T^{12} - 19180 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 - 312 T + 52972 T^{2} - 6909768 T^{3} + 710527206 T^{4} - 6909768 p^{2} T^{5} + 52972 p^{4} T^{6} - 312 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 68 T + 25546 T^{2} + 805520 T^{3} + 2932915 p T^{4} + 805520 p^{2} T^{5} + 25546 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.61508117284610578767249319534, −3.59407952345187560990045478610, −3.49771842239869160561286101006, −3.32110113727956984480117489679, −3.26473832329013798222392096743, −3.18003661423528055504375555462, −3.11704115904753306691675924406, −2.83893795070776727102242352719, −2.80603133206306845605260989378, −2.53320683191847750127347708194, −2.40590344529190265202791950219, −2.40090645648639848557854998537, −2.03671156391229720273808163512, −1.94618432263626450164782098846, −1.86830006350095114456484087911, −1.72018799968978064860432745360, −1.54693129803942924866378478296, −1.29196877589769092398566710375, −1.08523839958359691368717247596, −0.988435413613614603708136115045, −0.908135406409234137233289337103, −0.75387974284377392798530027722, −0.62856134703068543009087940889, −0.12513621718732863537364936728, −0.06387286715038366549294532381, 0.06387286715038366549294532381, 0.12513621718732863537364936728, 0.62856134703068543009087940889, 0.75387974284377392798530027722, 0.908135406409234137233289337103, 0.988435413613614603708136115045, 1.08523839958359691368717247596, 1.29196877589769092398566710375, 1.54693129803942924866378478296, 1.72018799968978064860432745360, 1.86830006350095114456484087911, 1.94618432263626450164782098846, 2.03671156391229720273808163512, 2.40090645648639848557854998537, 2.40590344529190265202791950219, 2.53320683191847750127347708194, 2.80603133206306845605260989378, 2.83893795070776727102242352719, 3.11704115904753306691675924406, 3.18003661423528055504375555462, 3.26473832329013798222392096743, 3.32110113727956984480117489679, 3.49771842239869160561286101006, 3.59407952345187560990045478610, 3.61508117284610578767249319534

Graph of the $Z$-function along the critical line

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