Properties

Label 16-1472e8-1.1-c1e8-0-0
Degree $16$
Conductor $2.204\times 10^{25}$
Sign $1$
Analytic cond. $3.64316\times 10^{8}$
Root an. cond. $3.42840$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 24·9-s − 56·49-s + 324·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 8·9-s − 8·49-s + 36·81-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 + 8·169-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(3.64316\times 10^{8}\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2535232564\)
\(L(\frac12)\) \(\approx\) \(0.2535232564\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( ( 1 + p T^{2} )^{4} \)
good3 \( ( 1 + p T^{2} )^{8} \)
5 \( ( 1 - 42 T^{4} + p^{4} T^{8} )^{2} \)
7 \( ( 1 + p T^{2} )^{8} \)
11 \( ( 1 + 150 T^{4} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - p T^{2} )^{8} \)
17 \( ( 1 - p T^{2} )^{8} \)
19 \( ( 1 + 630 T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - p T^{2} )^{8} \)
31 \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 1770 T^{4} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 810 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 + 1110 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 + 7350 T^{4} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 4470 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + p T^{2} )^{8} \)
83 \( ( 1 - 12810 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 - p T^{2} )^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.86435049453839051409941504765, −3.80727114607320931391896797420, −3.54278878327944923712326826428, −3.51543400009347514399929374051, −3.39727092282995342581525988359, −3.38226256236228159172567390728, −3.14800840907423645696040757902, −3.11386387690465826522731801250, −2.98040560144960501088618373655, −2.91559006285694047322890175204, −2.85076866599662300525967976877, −2.68949170469390789328204407063, −2.45119549870087173472430686598, −2.34683113271580161142778332825, −2.29499094838776145664633418840, −2.06025100583167401427033202001, −1.93704521459027060719711839930, −1.69949810424729057754076709054, −1.53735942489445607249577982380, −1.46408388628849698568588690651, −0.909859837434041623540232811268, −0.78545072692815569191357294212, −0.41440683688438305405515625671, −0.28558121446666688944813274362, −0.16841429039001183893974432430, 0.16841429039001183893974432430, 0.28558121446666688944813274362, 0.41440683688438305405515625671, 0.78545072692815569191357294212, 0.909859837434041623540232811268, 1.46408388628849698568588690651, 1.53735942489445607249577982380, 1.69949810424729057754076709054, 1.93704521459027060719711839930, 2.06025100583167401427033202001, 2.29499094838776145664633418840, 2.34683113271580161142778332825, 2.45119549870087173472430686598, 2.68949170469390789328204407063, 2.85076866599662300525967976877, 2.91559006285694047322890175204, 2.98040560144960501088618373655, 3.11386387690465826522731801250, 3.14800840907423645696040757902, 3.38226256236228159172567390728, 3.39727092282995342581525988359, 3.51543400009347514399929374051, 3.54278878327944923712326826428, 3.80727114607320931391896797420, 3.86435049453839051409941504765

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

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