↑

Properties

Label 16-1440e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.849\times 10^{25}$
Sign $1$
Analytic cond. $0.0711470$
Root an. cond. $0.847734$
Motivic weight $0$
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  + 2·25-s − 12·29-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯
L(s)  = 1  + 2·25-s − 12·29-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.0711470\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2705539095\)
\(L(\frac12)\) \(\approx\) \(0.2705539095\)
\(L(1)\) 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 \( 1 - T^{4} + T^{8} \)
5 \( ( 1 - T^{2} + T^{4} )^{2} \)
good7 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
17 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T^{2} )^{8} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T )^{8}( 1 + T )^{8} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
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

−4.20161972083372861273093438185, −4.16371728020370826332622017686, −4.05747414233073226335486455276, −3.84233585379498582247354340656, −3.70583320826861779149487464319, −3.63128867285484925002118961251, −3.53623442669649551367976186878, −3.45663528421374328759718544996, −3.41204353638198350752088867993, −3.19814834111105719778202090707, −2.99682199180946684505409853231, −2.97538249506154178524729505129, −2.65421908336412069359542945427, −2.52031186992905510502985418937, −2.33823166073465499462402561241, −2.15355109339415842354511625985, −2.11789376020234597892572942799, −2.02791327080258271498199240111, −1.77288544827439706424182656201, −1.66540134627859796854188377713, −1.53206816580416030233487271724, −1.42480268940059944557533779177, −1.17774173025382543167892218651, −0.77434470585115840678542513468, −0.26154232795915200742277493823, 0.26154232795915200742277493823, 0.77434470585115840678542513468, 1.17774173025382543167892218651, 1.42480268940059944557533779177, 1.53206816580416030233487271724, 1.66540134627859796854188377713, 1.77288544827439706424182656201, 2.02791327080258271498199240111, 2.11789376020234597892572942799, 2.15355109339415842354511625985, 2.33823166073465499462402561241, 2.52031186992905510502985418937, 2.65421908336412069359542945427, 2.97538249506154178524729505129, 2.99682199180946684505409853231, 3.19814834111105719778202090707, 3.41204353638198350752088867993, 3.45663528421374328759718544996, 3.53623442669649551367976186878, 3.63128867285484925002118961251, 3.70583320826861779149487464319, 3.84233585379498582247354340656, 4.05747414233073226335486455276, 4.16371728020370826332622017686, 4.20161972083372861273093438185

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

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