Properties

Label 16-440e8-1.1-c1e8-0-6
Degree $16$
Conductor $1.405\times 10^{21}$
Sign $1$
Analytic cond. $23218.7$
Root an. cond. $1.87441$
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  + 5·9-s + 8·11-s + 18·19-s − 25-s − 6·29-s − 30·31-s + 20·41-s + 19·49-s − 12·59-s + 58·61-s − 2·71-s + 40·79-s + 24·81-s − 42·89-s + 40·99-s − 24·101-s − 8·109-s + 36·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 5/3·9-s + 2.41·11-s + 4.12·19-s − 1/5·25-s − 1.11·29-s − 5.38·31-s + 3.12·41-s + 19/7·49-s − 1.56·59-s + 7.42·61-s − 0.237·71-s + 4.50·79-s + 8/3·81-s − 4.45·89-s + 4.02·99-s − 2.38·101-s − 0.766·109-s + 3.27·121-s − 0.536·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(23218.7\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.371968985\)
\(L(\frac12)\) \(\approx\) \(9.371968985\)
\(L(\frac{3}{2})\) 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 \( 1 + T^{2} + 6 T^{3} - 32 T^{4} + 6 p T^{5} + p^{2} T^{6} + p^{4} T^{8} \)
11 \( ( 1 - T )^{8} \)
good3 \( 1 - 5 T^{2} + T^{4} + 2 p T^{6} + 58 T^{8} + 2 p^{3} T^{10} + p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 19 T^{2} + 26 p T^{4} - 1773 T^{6} + 15282 T^{8} - 1773 p^{2} T^{10} + 26 p^{5} T^{12} - 19 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \)
17 \( 1 + T^{2} + 206 T^{4} - 7137 T^{6} - 52254 T^{8} - 7137 p^{2} T^{10} + 206 p^{4} T^{12} + p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 9 T + 56 T^{2} - 257 T^{3} + 1278 T^{4} - 257 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 30 T^{2} + 1369 T^{4} - 29258 T^{6} + 800540 T^{8} - 29258 p^{2} T^{10} + 1369 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 3 T + 78 T^{2} + 9 p T^{3} + 2874 T^{4} + 9 p^{2} T^{5} + 78 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 15 T + 167 T^{2} + 1312 T^{3} + 8448 T^{4} + 1312 p T^{5} + 167 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 161 T^{2} + 13001 T^{4} - 730714 T^{6} + 839458 p T^{8} - 730714 p^{2} T^{10} + 13001 p^{4} T^{12} - 161 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 10 T + 116 T^{2} - 654 T^{3} + 5126 T^{4} - 654 p T^{5} + 116 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 152 T^{2} + 12188 T^{4} - 643432 T^{6} + 28886950 T^{8} - 643432 p^{2} T^{10} + 12188 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 252 T^{2} + 31988 T^{4} - 2582916 T^{6} + 144448150 T^{8} - 2582916 p^{2} T^{10} + 31988 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 247 T^{2} + 31982 T^{4} - 2706865 T^{6} + 166655490 T^{8} - 2706865 p^{2} T^{10} + 31982 p^{4} T^{12} - 247 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 6 T + 189 T^{2} + 1138 T^{3} + 15308 T^{4} + 1138 p T^{5} + 189 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 29 T + 518 T^{2} - 6171 T^{3} + 55914 T^{4} - 6171 p T^{5} + 518 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 334 T^{2} + 56089 T^{4} - 6167034 T^{6} + 483745580 T^{8} - 6167034 p^{2} T^{10} + 56089 p^{4} T^{12} - 334 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + T + 147 T^{2} - 652 T^{3} + 9456 T^{4} - 652 p T^{5} + 147 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 356 T^{2} + 61124 T^{4} - 6797660 T^{6} + 562230966 T^{8} - 6797660 p^{2} T^{10} + 61124 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 20 T + 224 T^{2} - 868 T^{3} + 3454 T^{4} - 868 p T^{5} + 224 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 140 T^{2} + 14804 T^{4} - 1486228 T^{6} + 146071030 T^{8} - 1486228 p^{2} T^{10} + 14804 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 21 T + 477 T^{2} + 5682 T^{3} + 68718 T^{4} + 5682 p T^{5} + 477 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 466 T^{2} + 111905 T^{4} - 183522 p T^{6} + 2021503524 T^{8} - 183522 p^{3} T^{10} + 111905 p^{4} T^{12} - 466 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.90416015728241865315916024226, −4.82925200549877485150143948006, −4.54389908106623049397871124333, −4.53391231898670640540356479382, −4.04639869856319331383710435753, −4.01735024509585417568340446694, −4.00251110926346233435479728813, −3.97324939615025537803168580871, −3.79086759660357521935344798240, −3.72155750953396135761745437723, −3.47690279578739971815371405161, −3.36622194874735084846068990175, −3.36211292532851954184092710292, −2.79354476565679348158114182262, −2.75740899915465606808004691186, −2.67118591724459015046118865110, −2.23289473211796353211410446508, −2.16380909989744162391535422201, −1.92826624086521500288478528973, −1.82274532264421898656091915310, −1.39436105128766359100462172793, −1.38300258942287218742815154111, −1.07166490954719855280484516641, −0.74626573643413491233494025647, −0.70845767846003614932981050989, 0.70845767846003614932981050989, 0.74626573643413491233494025647, 1.07166490954719855280484516641, 1.38300258942287218742815154111, 1.39436105128766359100462172793, 1.82274532264421898656091915310, 1.92826624086521500288478528973, 2.16380909989744162391535422201, 2.23289473211796353211410446508, 2.67118591724459015046118865110, 2.75740899915465606808004691186, 2.79354476565679348158114182262, 3.36211292532851954184092710292, 3.36622194874735084846068990175, 3.47690279578739971815371405161, 3.72155750953396135761745437723, 3.79086759660357521935344798240, 3.97324939615025537803168580871, 4.00251110926346233435479728813, 4.01735024509585417568340446694, 4.04639869856319331383710435753, 4.53391231898670640540356479382, 4.54389908106623049397871124333, 4.82925200549877485150143948006, 4.90416015728241865315916024226

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

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