Properties

Label 32-960e16-1.1-c0e16-0-0
Degree $32$
Conductor $5.204\times 10^{47}$
Sign $1$
Analytic cond. $7.70643\times 10^{-6}$
Root an. cond. $0.692172$
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  − 16·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 251-s + 257-s + 263-s + ⋯
L(s)  = 1  − 16·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 251-s + 257-s + 263-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{96} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(7.70643\times 10^{-6}\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{96} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3035316476\)
\(L(\frac12)\) \(\approx\) \(0.3035316476\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{16} \)
3 \( 1 + T^{16} \)
5 \( 1 + T^{16} \)
good7 \( ( 1 + T^{8} )^{4} \)
11 \( ( 1 + T^{16} )^{2} \)
13 \( ( 1 + T^{16} )^{2} \)
17 \( ( 1 + T^{16} )^{2} \)
19 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
23 \( ( 1 + T^{16} )^{2} \)
29 \( ( 1 + T^{16} )^{2} \)
31 \( ( 1 + T^{8} )^{4} \)
37 \( ( 1 + T^{16} )^{2} \)
41 \( ( 1 + T^{8} )^{4} \)
43 \( ( 1 + T^{16} )^{2} \)
47 \( ( 1 + T^{16} )^{2} \)
53 \( ( 1 + T^{16} )^{2} \)
59 \( ( 1 + T^{16} )^{2} \)
61 \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \)
67 \( ( 1 + T^{16} )^{2} \)
71 \( ( 1 + T^{8} )^{4} \)
73 \( ( 1 + T^{8} )^{4} \)
79 \( ( 1 + T )^{16}( 1 + T^{2} )^{8} \)
83 \( ( 1 + T^{16} )^{2} \)
89 \( ( 1 + T^{8} )^{4} \)
97 \( ( 1 + T^{2} )^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.88382182842019981485235886192, −2.85660281815965038394540389152, −2.78847861580325377275382927568, −2.75969476923195816037271619409, −2.73920380934427429150708416734, −2.59884759174056316058842349841, −2.52641905046569448820966043719, −2.31813725068170469308568262804, −2.13349022541472722125959876556, −2.13318016414910592432695541879, −1.99970084154301319681594555516, −1.96821148694832773967519628178, −1.95826932727521786148354444771, −1.93930814312629708338747869783, −1.92591494565642177665614889968, −1.49029632519898002964560383913, −1.45832731178677814512245356744, −1.37550894749488483039028189351, −1.36720292957417180416800769734, −1.36507989036505711871277021616, −1.25831427561184842027981259474, −1.19654697643366367236649046076, −0.935864577919110723324970379796, −0.67176482383833027966779227335, −0.46847616662544562363956571751, 0.46847616662544562363956571751, 0.67176482383833027966779227335, 0.935864577919110723324970379796, 1.19654697643366367236649046076, 1.25831427561184842027981259474, 1.36507989036505711871277021616, 1.36720292957417180416800769734, 1.37550894749488483039028189351, 1.45832731178677814512245356744, 1.49029632519898002964560383913, 1.92591494565642177665614889968, 1.93930814312629708338747869783, 1.95826932727521786148354444771, 1.96821148694832773967519628178, 1.99970084154301319681594555516, 2.13318016414910592432695541879, 2.13349022541472722125959876556, 2.31813725068170469308568262804, 2.52641905046569448820966043719, 2.59884759174056316058842349841, 2.73920380934427429150708416734, 2.75969476923195816037271619409, 2.78847861580325377275382927568, 2.85660281815965038394540389152, 2.88382182842019981485235886192

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

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