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Properties

Label 16-7200e8-1.1-c1e8-0-9
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  − 8·31-s − 8·43-s + 28·49-s + 8·53-s + 24·67-s − 40·71-s − 16·79-s + 32·83-s + 32·107-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 60·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 1.43·31-s − 1.21·43-s + 4·49-s + 1.09·53-s + 2.93·67-s − 4.74·71-s − 1.80·79-s + 3.51·83-s + 3.09·107-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 + 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 + 0.0712·197-s + 0.0708·199-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

\(L(1)\) \(\approx\) \(18.27683131\)
\(L(\frac12)\) \(\approx\) \(18.27683131\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 p T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 2224 p^{2} T^{10} + 330 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
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 \( ( 1 + 30 T^{2} - 32 T^{3} + 451 T^{4} - 32 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 24968 p^{2} T^{10} + 1484 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
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 \( 1 - 56 T^{2} + 1324 T^{4} - 1592 p T^{6} + 1094310 T^{8} - 1592 p^{3} T^{10} + 1324 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 + 84 T^{2} - 128 T^{3} + 3542 T^{4} - 128 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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 \( ( 1 + 4 T + 58 T^{2} + 336 T^{3} + 3379 T^{4} + 336 p T^{5} + 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 1910872 p^{2} T^{10} + 26076 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 4 T + 92 T^{2} + 44 T^{3} + 3982 T^{4} + 44 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( ( 1 - 12 T + 154 T^{2} - 1520 T^{3} + 16755 T^{4} - 1520 p T^{5} + 154 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 1235336 p^{2} T^{10} + 15196 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 - 16 T + 276 T^{2} - 2576 T^{3} + 30070 T^{4} - 2576 p T^{5} + 276 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 11927760 p^{2} T^{10} + 80362 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.21316238706489439217635566331, −3.10379060866524906720683417268, −3.05273459215403327639841145379, −2.84430624785790529383025910874, −2.71005683391168263983038472802, −2.65412201750848334649850801330, −2.65061806973662721047348815099, −2.51180426547800018661815031044, −2.21258602951661320446171398283, −2.19548091083463118433549845672, −2.11298363362580211365559204388, −1.91494773700096691992132593128, −1.80948708724123551413574045673, −1.80844503233899297886087963571, −1.73511485638872602793777028120, −1.62668509660903625740424530744, −1.41136188090824498767855737261, −1.07171897113512233319299201976, −1.05918815912529860641636594603, −0.942917653022729256911512663616, −0.75930501405234971411314958227, −0.55980549081233861334611271896, −0.46618433199657005086950150046, −0.45750316198845612219206693297, −0.23025832874562496155719786865, 0.23025832874562496155719786865, 0.45750316198845612219206693297, 0.46618433199657005086950150046, 0.55980549081233861334611271896, 0.75930501405234971411314958227, 0.942917653022729256911512663616, 1.05918815912529860641636594603, 1.07171897113512233319299201976, 1.41136188090824498767855737261, 1.62668509660903625740424530744, 1.73511485638872602793777028120, 1.80844503233899297886087963571, 1.80948708724123551413574045673, 1.91494773700096691992132593128, 2.11298363362580211365559204388, 2.19548091083463118433549845672, 2.21258602951661320446171398283, 2.51180426547800018661815031044, 2.65061806973662721047348815099, 2.65412201750848334649850801330, 2.71005683391168263983038472802, 2.84430624785790529383025910874, 3.05273459215403327639841145379, 3.10379060866524906720683417268, 3.21316238706489439217635566331

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

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