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Properties

Label 32-935e16-1.1-c0e16-0-0
Degree $32$
Conductor $3.412\times 10^{47}$
Sign $1$
Analytic cond. $5.05242\times 10^{-6}$
Root an. cond. $0.683100$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·71-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  − 16·71-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(5^{16} \cdot 11^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(5.05242\times 10^{-6}\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 5^{16} \cdot 11^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09275525222\)
\(L(\frac12)\) \(\approx\) \(0.09275525222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T^{8} )^{2} \)
11 \( ( 1 + T^{8} )^{2} \)
17 \( 1 + T^{16} \)
good2 \( ( 1 + T^{16} )^{2} \)
3 \( ( 1 + T^{8} )^{4} \)
7 \( ( 1 + T^{16} )^{2} \)
13 \( ( 1 + T^{16} )^{2} \)
19 \( ( 1 + T^{4} )^{8} \)
23 \( ( 1 + T^{8} )^{4} \)
29 \( ( 1 + T^{8} )^{4} \)
31 \( ( 1 + T^{8} )^{4} \)
37 \( ( 1 + T^{8} )^{4} \)
41 \( ( 1 + T^{8} )^{4} \)
43 \( ( 1 + T^{16} )^{2} \)
47 \( ( 1 + T^{2} )^{16} \)
53 \( ( 1 + T^{4} )^{8} \)
59 \( ( 1 + T^{4} )^{8} \)
61 \( ( 1 + T^{8} )^{4} \)
67 \( ( 1 - T )^{16}( 1 + T )^{16} \)
71 \( ( 1 + T )^{16}( 1 + T^{4} )^{4} \)
73 \( ( 1 + T^{16} )^{2} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( ( 1 + T^{16} )^{2} \)
89 \( ( 1 + T^{8} )^{4} \)
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

−2.75030722863826068481841560735, −2.67510455642308582944582763121, −2.67159871907581966006794539468, −2.61775615446491222509812334654, −2.61286609232304785891014338220, −2.56906719944037018496704454424, −2.49522380334417336162324296746, −2.47739684351878657863758756924, −2.34452755926228797378424978907, −2.33714376125482916054723212878, −2.22263899775100979882562705132, −2.03795268364189739361279878548, −1.78012740753782642412542567225, −1.71998365660705014530523070954, −1.66738157359713449631823364755, −1.66073695437158975569520679988, −1.55727103053502811059764729118, −1.54930350855381957802519194816, −1.37121072053558722098846936822, −1.25636337042576625497100211905, −1.19955509521274285632157197641, −1.14575928188779777798592660633, −0.983746600126137151343462989505, −0.909467544581652256209172749295, −0.21444715745825830983443178467, 0.21444715745825830983443178467, 0.909467544581652256209172749295, 0.983746600126137151343462989505, 1.14575928188779777798592660633, 1.19955509521274285632157197641, 1.25636337042576625497100211905, 1.37121072053558722098846936822, 1.54930350855381957802519194816, 1.55727103053502811059764729118, 1.66073695437158975569520679988, 1.66738157359713449631823364755, 1.71998365660705014530523070954, 1.78012740753782642412542567225, 2.03795268364189739361279878548, 2.22263899775100979882562705132, 2.33714376125482916054723212878, 2.34452755926228797378424978907, 2.47739684351878657863758756924, 2.49522380334417336162324296746, 2.56906719944037018496704454424, 2.61286609232304785891014338220, 2.61775615446491222509812334654, 2.67159871907581966006794539468, 2.67510455642308582944582763121, 2.75030722863826068481841560735

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

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