Properties

Label 32-693e16-1.1-c0e16-0-0
Degree $32$
Conductor $2.830\times 10^{45}$
Sign $1$
Analytic cond. $4.19030\times 10^{-8}$
Root an. cond. $0.588091$
Motivic weight $0$
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  + 2·16-s + 4·25-s + 2·49-s − 8·67-s − 20·79-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 + ⋯
L(s)  = 1  + 2·16-s + 4·25-s + 2·49-s − 8·67-s − 20·79-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 7^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4.19030\times 10^{-8}\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{693} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2270370918\)
\(L(\frac12)\) \(\approx\) \(0.2270370918\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good2 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
53 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{8} \)
71 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
79 \( ( 1 + T )^{16}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
89 \( ( 1 + T^{2} )^{16} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.96516354527302468125286854457, −2.95567458311936212579182386401, −2.93546457510629599122432856361, −2.82611081045605333172500756675, −2.79757146720682771443586220879, −2.79345917471770236625237965966, −2.74640021440156624103945150881, −2.63805802925140126106760694773, −2.47461377835267084633176270288, −2.46170051947185904185253277059, −2.35405747009839158525114193277, −2.27602869291818794927641599057, −2.07150309078283174174464477932, −1.89201622697530961267569647771, −1.71686687436873811534454076603, −1.59401119483080453256658171939, −1.57709066657913199015786031238, −1.56068324131744474829727750197, −1.53608569331353593124517473024, −1.35212972536604054835604403879, −1.26000879937970853575463934527, −1.23470822574348972553277143840, −1.16817165119937264475905190244, −1.04958964635061490947976977415, −0.63545503605285065932662652634, 0.63545503605285065932662652634, 1.04958964635061490947976977415, 1.16817165119937264475905190244, 1.23470822574348972553277143840, 1.26000879937970853575463934527, 1.35212972536604054835604403879, 1.53608569331353593124517473024, 1.56068324131744474829727750197, 1.57709066657913199015786031238, 1.59401119483080453256658171939, 1.71686687436873811534454076603, 1.89201622697530961267569647771, 2.07150309078283174174464477932, 2.27602869291818794927641599057, 2.35405747009839158525114193277, 2.46170051947185904185253277059, 2.47461377835267084633176270288, 2.63805802925140126106760694773, 2.74640021440156624103945150881, 2.79345917471770236625237965966, 2.79757146720682771443586220879, 2.82611081045605333172500756675, 2.93546457510629599122432856361, 2.95567458311936212579182386401, 2.96516354527302468125286854457

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

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