Properties

Label 32-873e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.138\times 10^{47}$
Sign $1$
Analytic cond. $1.68555\times 10^{-6}$
Root an. cond. $0.660063$
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  − 16·79-s − 16·103-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 + ⋯
L(s)  = 1  − 16·79-s − 16·103-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 97^{16}\)
Sign: $1$
Analytic conductor: \(1.68555\times 10^{-6}\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{873} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 97^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07030885815\)
\(L(\frac12)\) \(\approx\) \(0.07030885815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.91259011959098119757099618194, −2.80782177712783763677778499972, −2.78438492725780323399008494488, −2.74930528932712349599109539927, −2.71594477581048078577143756092, −2.68894286356928013922568440408, −2.61053819148582862180905099096, −2.50836230081696021048207216122, −2.34220287757922849915631445119, −2.13663072865797545152561821103, −2.10709150590364125851547344004, −2.07391795022933394223123206073, −2.07197667636072683083826299033, −1.81914376239551131050008827839, −1.64198655457169020702685610126, −1.61794798135819418251357889077, −1.59625980175788511168599768024, −1.45400868270340261821843340853, −1.40097538325059712507504642898, −1.37977143531353131221302993029, −1.14183327764319155809840419064, −1.13281357564073279559878800395, −1.09589021186535098728036007809, −0.970883082900715143615229219429, −0.20354702442586224585966847381, 0.20354702442586224585966847381, 0.970883082900715143615229219429, 1.09589021186535098728036007809, 1.13281357564073279559878800395, 1.14183327764319155809840419064, 1.37977143531353131221302993029, 1.40097538325059712507504642898, 1.45400868270340261821843340853, 1.59625980175788511168599768024, 1.61794798135819418251357889077, 1.64198655457169020702685610126, 1.81914376239551131050008827839, 2.07197667636072683083826299033, 2.07391795022933394223123206073, 2.10709150590364125851547344004, 2.13663072865797545152561821103, 2.34220287757922849915631445119, 2.50836230081696021048207216122, 2.61053819148582862180905099096, 2.68894286356928013922568440408, 2.71594477581048078577143756092, 2.74930528932712349599109539927, 2.78438492725780323399008494488, 2.80782177712783763677778499972, 2.91259011959098119757099618194

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

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