Properties

Label 32-980e16-1.1-c0e16-0-0
Degree $32$
Conductor $7.238\times 10^{47}$
Sign $1$
Analytic cond. $1.07184\times 10^{-5}$
Root an. cond. $0.699345$
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 − 8·53-s + 4·81-s − 16·113-s − 8·121-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 + ⋯
L(s)  = 1  + 2·16-s − 8·53-s + 4·81-s − 16·113-s − 8·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 5^{16} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(1.07184\times 10^{-5}\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1540889744\)
\(L(\frac12)\) \(\approx\) \(0.1540889744\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.90908789726601974994207346512, −2.87879930550531657527706266394, −2.81285987596997729614699326455, −2.67855138154227589989887985242, −2.57688574310364587128843940915, −2.47895911302942702865657782955, −2.38780350328171679006700652100, −2.32359592202046439146272450793, −2.31622087899353581938902121366, −2.26745984249529510201006063058, −2.25334923901345136494817088892, −1.91222798708511541510137391799, −1.90046720803874689887790378417, −1.80173767710441374026707575090, −1.70160829947482711061508608904, −1.52419860039426188853104628960, −1.43215556003379197628671877211, −1.37713985780495308752362656855, −1.35823294481175125934541260987, −1.26534666742680647653959682610, −1.25477056353931840363894589746, −1.22572677767808113077800538271, −0.966980744804699731186136184365, −0.886067886780049811381084181155, −0.25224155225769042694524968774, 0.25224155225769042694524968774, 0.886067886780049811381084181155, 0.966980744804699731186136184365, 1.22572677767808113077800538271, 1.25477056353931840363894589746, 1.26534666742680647653959682610, 1.35823294481175125934541260987, 1.37713985780495308752362656855, 1.43215556003379197628671877211, 1.52419860039426188853104628960, 1.70160829947482711061508608904, 1.80173767710441374026707575090, 1.90046720803874689887790378417, 1.91222798708511541510137391799, 2.25334923901345136494817088892, 2.26745984249529510201006063058, 2.31622087899353581938902121366, 2.32359592202046439146272450793, 2.38780350328171679006700652100, 2.47895911302942702865657782955, 2.57688574310364587128843940915, 2.67855138154227589989887985242, 2.81285987596997729614699326455, 2.87879930550531657527706266394, 2.90908789726601974994207346512

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

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