↑

Properties

Label 32-60e32-1.1-c0e16-0-0
Degree $32$
Conductor $7.959\times 10^{56}$
Sign $1$
Analytic cond. $11785.6$
Root an. cond. $1.34038$
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  − 4·13-s + 4·37-s + 4·73-s + 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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  − 4·13-s + 4·37-s + 4·73-s + 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 3^{32} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(11785.6\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 3^{32} \cdot 5^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.911170859\)
\(L(\frac12)\) \(\approx\) \(2.911170859\)
\(L(1)\) 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 \( 1 \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good7 \( ( 1 + T^{4} )^{8} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
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 \( ( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
41 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
43 \( ( 1 + T^{4} )^{8} \)
47 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
53 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
89 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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

−2.28560742611870771667674937585, −2.24611879573413677463822039234, −2.13895075802270648919704488491, −2.13065084744309204465015778700, −2.08896588780027101286253778515, −2.01004254692297942261471047378, −2.00055932447720786118785168726, −1.84821757138007885464556408404, −1.79984496134697849913559271708, −1.71343137067215097141543287452, −1.64145523951003654045980133972, −1.64030205045955792117926725812, −1.58374008495254763340833843470, −1.33933212865558841797143027348, −1.25494633908653020092728810319, −1.12747058406160642558819075345, −1.08674627387388856452880654864, −1.07578613619746868521631991607, −1.04346522296761098263226343852, −0.840549776385775673521892692054, −0.73293158763106898779783793541, −0.69900473873621712118897565285, −0.47229230530386823641710718887, −0.45349841264741748273803302251, −0.39373198153570822192600322678, 0.39373198153570822192600322678, 0.45349841264741748273803302251, 0.47229230530386823641710718887, 0.69900473873621712118897565285, 0.73293158763106898779783793541, 0.840549776385775673521892692054, 1.04346522296761098263226343852, 1.07578613619746868521631991607, 1.08674627387388856452880654864, 1.12747058406160642558819075345, 1.25494633908653020092728810319, 1.33933212865558841797143027348, 1.58374008495254763340833843470, 1.64030205045955792117926725812, 1.64145523951003654045980133972, 1.71343137067215097141543287452, 1.79984496134697849913559271708, 1.84821757138007885464556408404, 2.00055932447720786118785168726, 2.01004254692297942261471047378, 2.08896588780027101286253778515, 2.13065084744309204465015778700, 2.13895075802270648919704488491, 2.24611879573413677463822039234, 2.28560742611870771667674937585

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

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