↑

Properties

Label 16-9800e8-1.1-c1e8-0-1
Degree $16$
Conductor $8.508\times 10^{31}$
Sign $1$
Analytic cond. $1.40612\times 10^{15}$
Root an. cond. $8.84609$
Motivic weight $1$
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  − 6·9-s + 4·11-s + 4·23-s + 8·29-s − 16·43-s + 28·53-s − 40·67-s + 8·71-s + 20·79-s + 23·81-s − 24·99-s − 76·107-s + 12·109-s + 32·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 66·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·9-s + 1.20·11-s + 0.834·23-s + 1.48·29-s − 2.43·43-s + 3.84·53-s − 4.88·67-s + 0.949·71-s + 2.25·79-s + 23/9·81-s − 2.41·99-s − 7.34·107-s + 1.14·109-s + 3.01·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.40612\times 10^{15}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.911702068\)
\(L(\frac12)\) \(\approx\) \(7.911702068\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 p T^{2} + 13 T^{4} + 64 T^{6} + 316 T^{8} + 64 p^{2} T^{10} + 13 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 2 T + 6 T^{2} - 4 T^{3} + 147 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 66 T^{2} + 2108 T^{4} + 44094 T^{6} + 666886 T^{8} + 44094 p^{2} T^{10} + 2108 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 52 T^{2} + 1613 T^{4} + 36440 T^{6} + 690556 T^{8} + 36440 p^{2} T^{10} + 1613 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 8 T^{2} - 135 T^{4} + 5888 T^{6} + 34928 T^{8} + 5888 p^{2} T^{10} - 135 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 2 T + 45 T^{2} + 18 T^{3} + 1004 T^{4} + 18 p T^{5} + 45 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 4 T + 31 T^{2} - 20 T^{3} - 176 T^{4} - 20 p T^{5} + 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 130 T^{2} + 8772 T^{4} + 418766 T^{6} + 15006422 T^{8} + 418766 p^{2} T^{10} + 8772 p^{4} T^{12} + 130 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 + 43 T^{2} + 92 T^{3} + 932 T^{4} + 92 p T^{5} + 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( 1 + 190 T^{2} + 17621 T^{4} + 1097480 T^{6} + 51314980 T^{8} + 1097480 p^{2} T^{10} + 17621 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 8 T + 91 T^{2} + 552 T^{3} + 4028 T^{4} + 552 p T^{5} + 91 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 28 T^{2} + 6184 T^{4} - 76596 T^{6} + 16621838 T^{8} - 76596 p^{2} T^{10} + 6184 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 14 T + 208 T^{2} - 1906 T^{3} + 16942 T^{4} - 1906 p T^{5} + 208 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 190 T^{2} + 17400 T^{4} + 1425002 T^{6} + 100823438 T^{8} + 1425002 p^{2} T^{10} + 17400 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 + 266 T^{2} + 36236 T^{4} + 3305206 T^{6} + 227629030 T^{8} + 3305206 p^{2} T^{10} + 36236 p^{4} T^{12} + 266 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 + 20 T + 346 T^{2} + 3768 T^{3} + 36179 T^{4} + 3768 p T^{5} + 346 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 4 T + 179 T^{2} - 408 T^{3} + 14996 T^{4} - 408 p T^{5} + 179 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 270 T^{2} + 33397 T^{4} + 3003592 T^{6} + 235852580 T^{8} + 3003592 p^{2} T^{10} + 33397 p^{4} T^{12} + 270 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 10 T + 137 T^{2} + 182 T^{3} + 1440 T^{4} + 182 p T^{5} + 137 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 486 T^{2} + 114541 T^{4} + 16838816 T^{6} + 1678490780 T^{8} + 16838816 p^{2} T^{10} + 114541 p^{4} T^{12} + 486 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 270 T^{2} + 45653 T^{4} + 5475384 T^{6} + 548443204 T^{8} + 5475384 p^{2} T^{10} + 45653 p^{4} T^{12} + 270 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 + 390 T^{2} + 66256 T^{4} + 7045522 T^{6} + 654890654 T^{8} + 7045522 p^{2} T^{10} + 66256 p^{4} T^{12} + 390 p^{6} T^{14} + p^{8} T^{16} \)
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

−3.15779035140609046478807211814, −2.81375400268842282679264865380, −2.78920381641755308698610776650, −2.72409899527935005540046810409, −2.69668013211283358781646745110, −2.64603928689946519957621014965, −2.62361818873995847681930136694, −2.47581158124070514955862927699, −2.36294548094504440341434263260, −1.93695456670049236349803375876, −1.92384861008077836865408261823, −1.77932756151497985750526014726, −1.74567696009728654517620965700, −1.72283341810739283731252551178, −1.62708485751852355692798297069, −1.61484150241056364349954039394, −1.31470994525384925373670598701, −1.09544087864346517638391291206, −0.940200093283162898902412946913, −0.76045461591554717823154627777, −0.65902949450682475223789816513, −0.64282791611332590869100471420, −0.63257193507613398915665298263, −0.21701236098198348692272855107, −0.19726993159930522019396068275, 0.19726993159930522019396068275, 0.21701236098198348692272855107, 0.63257193507613398915665298263, 0.64282791611332590869100471420, 0.65902949450682475223789816513, 0.76045461591554717823154627777, 0.940200093283162898902412946913, 1.09544087864346517638391291206, 1.31470994525384925373670598701, 1.61484150241056364349954039394, 1.62708485751852355692798297069, 1.72283341810739283731252551178, 1.74567696009728654517620965700, 1.77932756151497985750526014726, 1.92384861008077836865408261823, 1.93695456670049236349803375876, 2.36294548094504440341434263260, 2.47581158124070514955862927699, 2.62361818873995847681930136694, 2.64603928689946519957621014965, 2.69668013211283358781646745110, 2.72409899527935005540046810409, 2.78920381641755308698610776650, 2.81375400268842282679264865380, 3.15779035140609046478807211814

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

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