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Properties

Label 16-2400e8-1.1-c1e8-0-10
Degree $16$
Conductor $1.101\times 10^{27}$
Sign $1$
Analytic cond. $1.81931\times 10^{10}$
Root an. cond. $4.37768$
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 + 8·23-s − 8·31-s + 4·49-s − 32·63-s + 40·71-s + 16·73-s + 16·79-s + 10·81-s + 8·97-s + 64·103-s + 32·113-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 60·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 3.02·7-s − 4/3·9-s + 1.66·23-s − 1.43·31-s + 4/7·49-s − 4.03·63-s + 4.74·71-s + 1.87·73-s + 1.80·79-s + 10/9·81-s + 0.812·97-s + 6.30·103-s + 3.01·113-s + 5.09·121-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 + 0.0798·157-s + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 4.61·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.81931\times 10^{10}\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.33052067\)
\(L(\frac12)\) \(\approx\) \(20.33052067\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{4} \)
5 \( 1 \)
good7 \( ( 1 - 4 T + 22 T^{2} - 72 T^{3} + 211 T^{4} - 72 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 29896 p^{2} T^{10} + 1612 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 60 T^{2} + 1802 T^{4} - 36176 T^{6} + 538099 T^{8} - 36176 p^{2} T^{10} + 1802 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 28 T^{2} - 104 T^{3} + 350 T^{4} - 104 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 35120 p^{2} T^{10} + 1546 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 4 T + 36 T^{2} - 124 T^{3} + 510 T^{4} - 124 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 171048 p^{2} T^{10} + 4780 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 168 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 168 T^{2} + 14140 T^{4} - 808664 T^{6} + 34400998 T^{8} - 808664 p^{2} T^{10} + 14140 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 100 T^{2} + 56 T^{3} + 126 p T^{4} + 56 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 - 100 T^{2} + 7434 T^{4} - 377968 T^{6} + 18442035 T^{8} - 377968 p^{2} T^{10} + 7434 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 116 T^{2} - 256 T^{3} + 6310 T^{4} - 256 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 168 T^{2} + 16780 T^{4} - 1266264 T^{6} + 74218758 T^{8} - 1266264 p^{2} T^{10} + 16780 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 112984 p^{2} T^{10} + 2364 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 2942672 p^{2} T^{10} + 32650 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 164 T^{2} + 20746 T^{4} - 1788592 T^{6} + 137741171 T^{8} - 1788592 p^{2} T^{10} + 20746 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 4188 p T^{5} + 380 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 8 T + 124 T^{2} - 888 T^{3} + 7014 T^{4} - 888 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 8 T + 132 T^{2} - 1032 T^{3} + 16454 T^{4} - 1032 p T^{5} + 132 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 296 T^{2} + 53884 T^{4} - 6923736 T^{6} + 654380710 T^{8} - 6923736 p^{2} T^{10} + 53884 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 132 T^{2} + 64 T^{3} + 18534 T^{4} + 64 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 4 T + 186 T^{2} - 816 T^{3} + 26147 T^{4} - 816 p T^{5} + 186 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.61103367400549340136668523370, −3.61015482947213805480895221562, −3.36920201798437275889921775531, −3.29920575347050513103576820390, −3.28378868875170002378318391871, −3.20900945751861459016632135891, −3.18349894267803782683417213935, −3.02697639235422243873275710958, −2.65313824206539423346189608023, −2.64524344529039897094731960570, −2.28980633632573367596766472322, −2.28794147385858847928061816070, −2.18103944734026977771779900726, −2.09316875225004068097589727520, −1.86854674188397311144412374046, −1.86075366450334741100665197784, −1.73738405749235888007605225903, −1.71148294393198073971123310647, −1.41185093504927927469427605674, −1.02920738849212541195631510185, −1.00762980725013027862436398257, −0.77229503198967958741467975438, −0.66538688625100692452211937851, −0.55527370520430703048686584884, −0.30240438525546169422506466893, 0.30240438525546169422506466893, 0.55527370520430703048686584884, 0.66538688625100692452211937851, 0.77229503198967958741467975438, 1.00762980725013027862436398257, 1.02920738849212541195631510185, 1.41185093504927927469427605674, 1.71148294393198073971123310647, 1.73738405749235888007605225903, 1.86075366450334741100665197784, 1.86854674188397311144412374046, 2.09316875225004068097589727520, 2.18103944734026977771779900726, 2.28794147385858847928061816070, 2.28980633632573367596766472322, 2.64524344529039897094731960570, 2.65313824206539423346189608023, 3.02697639235422243873275710958, 3.18349894267803782683417213935, 3.20900945751861459016632135891, 3.28378868875170002378318391871, 3.29920575347050513103576820390, 3.36920201798437275889921775531, 3.61015482947213805480895221562, 3.61103367400549340136668523370

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

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