Properties

Label 32-820e16-1.1-c0e16-0-0
Degree $32$
Conductor $4.179\times 10^{46}$
Sign $1$
Analytic cond. $6.18779\times 10^{-7}$
Root an. cond. $0.639713$
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  + 2·4-s − 4·13-s + 16-s + 4·17-s − 4·29-s − 8·52-s + 4·53-s − 4·61-s + 8·68-s − 4·109-s − 8·116-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·4-s − 4·13-s + 16-s + 4·17-s − 4·29-s − 8·52-s + 4·53-s − 4·61-s + 8·68-s − 4·109-s − 8·116-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 5^{16} \cdot 41^{16}\)
Sign: $1$
Analytic conductor: \(6.18779\times 10^{-7}\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{820} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 5^{16} \cdot 41^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3672164935\)
\(L(\frac12)\) \(\approx\) \(0.3672164935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
good3 \( ( 1 + T^{8} )^{4} \)
7 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
11 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
19 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
23 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
37 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
47 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
71 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( ( 1 + T^{4} )^{8} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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

−2.94166812989547524497704088524, −2.84569026988039599268067648681, −2.82787762865579891425296423490, −2.77834122154358238460902753998, −2.63438486039748067285325631525, −2.61356029375699513854250328646, −2.50244897645930169507999669769, −2.48073275708322022828667356033, −2.35211641019870796640135412539, −2.26198746414356185445250173190, −2.21494520710589251224715152495, −2.09036363192310956360147522987, −2.04988732026400040769542014670, −1.92236376821973836403390783812, −1.87359634500319362106795455995, −1.81922715996155668109600019535, −1.52544442603589546066140491231, −1.50376312711409786448449027056, −1.48844056700146986365231632010, −1.41154215384679905759313621866, −1.34877257796171283199254093700, −1.04689021898343795788012576841, −1.00929201697721347311449467585, −0.924202550874431193811352940337, −0.52630451104177643059474481458, 0.52630451104177643059474481458, 0.924202550874431193811352940337, 1.00929201697721347311449467585, 1.04689021898343795788012576841, 1.34877257796171283199254093700, 1.41154215384679905759313621866, 1.48844056700146986365231632010, 1.50376312711409786448449027056, 1.52544442603589546066140491231, 1.81922715996155668109600019535, 1.87359634500319362106795455995, 1.92236376821973836403390783812, 2.04988732026400040769542014670, 2.09036363192310956360147522987, 2.21494520710589251224715152495, 2.26198746414356185445250173190, 2.35211641019870796640135412539, 2.48073275708322022828667356033, 2.50244897645930169507999669769, 2.61356029375699513854250328646, 2.63438486039748067285325631525, 2.77834122154358238460902753998, 2.82787762865579891425296423490, 2.84569026988039599268067648681, 2.94166812989547524497704088524

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

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