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Properties

Label 16-672e8-1.1-c1e8-0-9
Degree $16$
Conductor $4.159\times 10^{22}$
Sign $1$
Analytic cond. $687339.$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 4·9-s + 4·17-s − 12·23-s + 8·25-s − 8·31-s − 4·41-s + 36·49-s + 32·63-s + 28·71-s − 8·73-s + 40·79-s + 10·81-s + 20·89-s + 40·97-s − 8·103-s − 8·113-s − 32·119-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + ⋯
L(s)  = 1  − 3.02·7-s − 4/3·9-s + 0.970·17-s − 2.50·23-s + 8/5·25-s − 1.43·31-s − 0.624·41-s + 36/7·49-s + 4.03·63-s + 3.32·71-s − 0.936·73-s + 4.50·79-s + 10/9·81-s + 2.11·89-s + 4.06·97-s − 0.788·103-s − 0.752·113-s − 2.93·119-s + 2.18·121-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 − 1.29·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(687339.\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.467698335\)
\(L(\frac12)\) \(\approx\) \(2.467698335\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{2} )^{4} \)
7 \( ( 1 + T )^{8} \)
good5 \( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 168 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 8312 p^{2} T^{10} + 592 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 26704 p^{2} T^{10} + 1340 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 70 p T^{5} + 38 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 42432 p^{2} T^{10} + 2012 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 334 p T^{5} + 74 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 34928 p^{2} T^{10} + 2876 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 244 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1175872 p^{2} T^{10} + 18716 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 2 T + 134 T^{2} + 214 T^{3} + 7618 T^{4} + 214 p T^{5} + 134 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1026520 p^{2} T^{10} + 15676 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 128 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 5027376 p^{2} T^{10} + 52604 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \)
61 \( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 3030896 p^{2} T^{10} + 35516 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 6783272 p^{2} T^{10} + 59996 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 1686 p T^{5} + 194 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 580 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 20 T + 340 T^{2} - 3972 T^{3} + 38678 T^{4} - 3972 p T^{5} + 340 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 248 T^{2} + 532 p T^{4} - 5204488 T^{6} + 499369126 T^{8} - 5204488 p^{2} T^{10} + 532 p^{5} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 10 T + 198 T^{2} - 494 T^{3} + 13122 T^{4} - 494 p T^{5} + 198 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 2572 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.49016462901033389686382288489, −4.34742679918832510305077508914, −4.08042045884049859012933667854, −4.06105053458875008439726220617, −4.05164891612955381973817877322, −4.00408799469641709478310877918, −3.53302156970907584264258685596, −3.45050967869519113768929490690, −3.40989650807353022230008631612, −3.35151951317659564134683822012, −3.23297899597053526629589846514, −3.06162852962262700032813783632, −2.97535289615267483865543556691, −2.73088562185847073874759392351, −2.55423358264363743807354881153, −2.25629264749389193187153452818, −2.19521221554476824168278841791, −2.09570872080927239266768447886, −1.90183814289511450978458086845, −1.67812458116808409065748220128, −1.49236221692041960328235589386, −0.813460356922776492031875778548, −0.65598735063407581382633387420, −0.55789665340972688815260148631, −0.44227034401757427938518163838, 0.44227034401757427938518163838, 0.55789665340972688815260148631, 0.65598735063407581382633387420, 0.813460356922776492031875778548, 1.49236221692041960328235589386, 1.67812458116808409065748220128, 1.90183814289511450978458086845, 2.09570872080927239266768447886, 2.19521221554476824168278841791, 2.25629264749389193187153452818, 2.55423358264363743807354881153, 2.73088562185847073874759392351, 2.97535289615267483865543556691, 3.06162852962262700032813783632, 3.23297899597053526629589846514, 3.35151951317659564134683822012, 3.40989650807353022230008631612, 3.45050967869519113768929490690, 3.53302156970907584264258685596, 4.00408799469641709478310877918, 4.05164891612955381973817877322, 4.06105053458875008439726220617, 4.08042045884049859012933667854, 4.34742679918832510305077508914, 4.49016462901033389686382288489

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

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