Properties

Label 32-896e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.726\times 10^{47}$
Sign $1$
Analytic cond. $2.55530\times 10^{-6}$
Root an. cond. $0.668701$
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  − 16·107-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  − 16·107-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{112} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.55530\times 10^{-6}\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{896} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{112} \cdot 7^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2591745051\)
\(L(\frac12)\) \(\approx\) \(0.2591745051\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.85731327921526230397213453086, −2.83380656414832649389303683524, −2.77909637376431326714344090739, −2.59719565561349382622181878261, −2.56912683068754057524288460001, −2.56039348865748451892362559962, −2.50362998171137964668882483268, −2.48999041705503352304867473775, −2.46056788166653303947064380337, −2.43506151330254005419652350088, −2.11760821690747758599519481532, −2.05441656444153624065452811060, −1.96008175230361759246814383679, −1.72384267404992987022692851345, −1.71186877923010154005933932169, −1.59719120759213681589720411310, −1.51033176330278371140291600663, −1.50682452866242756158153507527, −1.45448663612628308723069310551, −1.35333550966473036527470653453, −1.27951006728395171995675577235, −1.06235904301495902466601847447, −0.952617870113289477444173460893, −0.848078503978862697565233712322, −0.43712567007360193083461238717, 0.43712567007360193083461238717, 0.848078503978862697565233712322, 0.952617870113289477444173460893, 1.06235904301495902466601847447, 1.27951006728395171995675577235, 1.35333550966473036527470653453, 1.45448663612628308723069310551, 1.50682452866242756158153507527, 1.51033176330278371140291600663, 1.59719120759213681589720411310, 1.71186877923010154005933932169, 1.72384267404992987022692851345, 1.96008175230361759246814383679, 2.05441656444153624065452811060, 2.11760821690747758599519481532, 2.43506151330254005419652350088, 2.46056788166653303947064380337, 2.48999041705503352304867473775, 2.50362998171137964668882483268, 2.56039348865748451892362559962, 2.56912683068754057524288460001, 2.59719565561349382622181878261, 2.77909637376431326714344090739, 2.83380656414832649389303683524, 2.85731327921526230397213453086

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

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