↑

Properties

Label 16-2100e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.782\times 10^{26}$
Sign $1$
Analytic cond. $1.45549$
Root an. cond. $1.02373$
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  + 4-s − 2·5-s + 9-s − 2·20-s + 25-s + 36-s − 6·41-s − 2·45-s − 4·49-s − 6·89-s + 100-s − 4·101-s + 6·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4-s − 2·5-s + 9-s − 2·20-s + 25-s + 36-s − 6·41-s − 2·45-s − 4·49-s − 6·89-s + 100-s − 4·101-s + 6·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 2·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.45549\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01367856931\)
\(L(\frac12)\) \(\approx\) \(0.01367856931\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 + T^{2} )^{4} \)
good11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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

−3.96214078708117477826473095440, −3.84749342753988130453491710374, −3.83106107883716734780876074238, −3.82746146063388460990580844279, −3.36743834191621573361441871826, −3.33753182949423586461701300382, −3.27156175144351648948417349764, −3.20329773762723739386959731091, −3.18401463375064432373620897716, −3.13033624427799534235972934436, −2.97209383755639217131002790291, −2.74542988332214347845112101028, −2.53753935630040646622250177536, −2.20189881270025192560667764143, −2.17572183965202336920081424447, −2.17497270396485218828777088569, −2.09692845632203604033741825466, −1.78745708450533201934691814883, −1.54255066701100172161844713261, −1.43805573376861536480324275920, −1.43601025651677027001863053681, −1.39883973921483034685289165994, −1.04519321532182794161959334727, −0.66408687092048804983442662841, −0.04419857125729080647102953214, 0.04419857125729080647102953214, 0.66408687092048804983442662841, 1.04519321532182794161959334727, 1.39883973921483034685289165994, 1.43601025651677027001863053681, 1.43805573376861536480324275920, 1.54255066701100172161844713261, 1.78745708450533201934691814883, 2.09692845632203604033741825466, 2.17497270396485218828777088569, 2.17572183965202336920081424447, 2.20189881270025192560667764143, 2.53753935630040646622250177536, 2.74542988332214347845112101028, 2.97209383755639217131002790291, 3.13033624427799534235972934436, 3.18401463375064432373620897716, 3.20329773762723739386959731091, 3.27156175144351648948417349764, 3.33753182949423586461701300382, 3.36743834191621573361441871826, 3.82746146063388460990580844279, 3.83106107883716734780876074238, 3.84749342753988130453491710374, 3.96214078708117477826473095440

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

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