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Properties

Label 16-7200e8-1.1-c1e8-0-0
Degree $16$
Conductor $7.222\times 10^{30}$
Sign $1$
Analytic cond. $1.19364\times 10^{14}$
Root an. cond. $7.58236$
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 − 8·23-s − 8·31-s + 4·49-s − 40·71-s + 16·73-s + 16·79-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 + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 3.02·7-s − 1.66·23-s − 1.43·31-s + 4/7·49-s − 4.74·71-s + 1.87·73-s + 1.80·79-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 + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.19364\times 10^{14}\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9409148491\)
\(L(\frac12)\) \(\approx\) \(0.9409148491\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−3.23719427830904584445023082071, −3.18026329889480579752784271369, −3.08938476498760416432755885613, −2.84248313181945500106284613810, −2.80787782661585174411369540610, −2.57177943046635331029932758941, −2.48005389261664018262695053504, −2.33717803545747948818503578642, −2.26058805875609664717339219943, −2.19661457596192727933877113077, −2.10146928757475369297659557947, −1.89875542745335330956931432761, −1.75342076538122193436632351550, −1.74473851311169200988609586684, −1.72248972396766007845436634087, −1.62416351400831302676995478407, −1.31777560284663421456480834280, −1.31219239643558366727528287757, −1.15096389018395017848759825043, −1.11064826137138620641932097530, −0.74739031576930957479118405651, −0.62899057023146097729355336372, −0.49818267585777766670505006771, −0.32609881815165167089961669601, −0.04893566805976089714485649191, 0.04893566805976089714485649191, 0.32609881815165167089961669601, 0.49818267585777766670505006771, 0.62899057023146097729355336372, 0.74739031576930957479118405651, 1.11064826137138620641932097530, 1.15096389018395017848759825043, 1.31219239643558366727528287757, 1.31777560284663421456480834280, 1.62416351400831302676995478407, 1.72248972396766007845436634087, 1.74473851311169200988609586684, 1.75342076538122193436632351550, 1.89875542745335330956931432761, 2.10146928757475369297659557947, 2.19661457596192727933877113077, 2.26058805875609664717339219943, 2.33717803545747948818503578642, 2.48005389261664018262695053504, 2.57177943046635331029932758941, 2.80787782661585174411369540610, 2.84248313181945500106284613810, 3.08938476498760416432755885613, 3.18026329889480579752784271369, 3.23719427830904584445023082071

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

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