Properties

Label 32-1067e16-1.1-c0e16-0-0
Degree $32$
Conductor $2.822\times 10^{48}$
Sign $1$
Analytic cond. $4.17969\times 10^{-5}$
Root an. cond. $0.729727$
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  + 8·31-s − 16·59-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 + 2·256-s + ⋯
L(s)  = 1  + 8·31-s − 16·59-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 + 2·256-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(11^{16} \cdot 97^{16}\)
Sign: $1$
Analytic conductor: \(4.17969\times 10^{-5}\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1067} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 11^{16} \cdot 97^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4844816548\)
\(L(\frac12)\) \(\approx\) \(0.4844816548\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T^{8} + T^{16} \)
97 \( 1 - T^{8} + T^{16} \)
good2 \( ( 1 - T^{8} + T^{16} )^{2} \)
3 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
5 \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \)
7 \( 1 - T^{16} + T^{32} \)
13 \( 1 - T^{16} + T^{32} \)
17 \( 1 - T^{16} + T^{32} \)
19 \( ( 1 + T^{16} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \)
29 \( 1 - T^{16} + T^{32} \)
31 \( ( 1 - T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \)
37 \( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
41 \( 1 - T^{16} + T^{32} \)
43 \( ( 1 - T^{4} + T^{8} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} )^{4} \)
53 \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{4} )^{4} \)
59 \( ( 1 + T )^{16}( 1 - T^{8} + T^{16} ) \)
61 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
71 \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \)
73 \( ( 1 - T^{4} + T^{8} )^{4} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( 1 - T^{16} + T^{32} \)
89 \( ( 1 + T^{8} )^{4} \)
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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

−2.89933683124853378567325164702, −2.88346183674096639410547543106, −2.84739695623538615850853776995, −2.79402747935443441209340765618, −2.58273208519936604419061499253, −2.51967165577021666062947054579, −2.38034685875807529743675520512, −2.35386346360254385633552007746, −2.15701465607510133754016963113, −2.12547522138973401958783025083, −2.05354353123625424157088121425, −1.95563620545801859067888576266, −1.93418041285458702402076523709, −1.79911545265745522051768294292, −1.58486966153799464944201455543, −1.52898247956457764245526933805, −1.46953938802172559578475646391, −1.41784092456366239893224405643, −1.33365117729265036454146320398, −1.28629528881719978592292595901, −1.13133748402568996364604023283, −1.12385688145111260477879424940, −0.918286163133349491708507931673, −0.69805277024854729059933214646, −0.48722567172832297592643217364, 0.48722567172832297592643217364, 0.69805277024854729059933214646, 0.918286163133349491708507931673, 1.12385688145111260477879424940, 1.13133748402568996364604023283, 1.28629528881719978592292595901, 1.33365117729265036454146320398, 1.41784092456366239893224405643, 1.46953938802172559578475646391, 1.52898247956457764245526933805, 1.58486966153799464944201455543, 1.79911545265745522051768294292, 1.93418041285458702402076523709, 1.95563620545801859067888576266, 2.05354353123625424157088121425, 2.12547522138973401958783025083, 2.15701465607510133754016963113, 2.35386346360254385633552007746, 2.38034685875807529743675520512, 2.51967165577021666062947054579, 2.58273208519936604419061499253, 2.79402747935443441209340765618, 2.84739695623538615850853776995, 2.88346183674096639410547543106, 2.89933683124853378567325164702

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

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