Properties

Label 32-1045e16-1.1-c0e16-0-0
Degree $32$
Conductor $2.022\times 10^{48}$
Sign $1$
Analytic cond. $2.99484\times 10^{-5}$
Root an. cond. $0.722165$
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  + 4·5-s + 2·16-s + 4·19-s + 6·25-s − 4·49-s + 8·80-s + 2·81-s + 16·95-s + 2·121-s + 4·125-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 + ⋯
L(s)  = 1  + 4·5-s + 2·16-s + 4·19-s + 6·25-s − 4·49-s + 8·80-s + 2·81-s + 16·95-s + 2·121-s + 4·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(5^{16} \cdot 11^{16} \cdot 19^{16}\)
Sign: $1$
Analytic conductor: \(2.99484\times 10^{-5}\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1045} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 5^{16} \cdot 11^{16} \cdot 19^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.968964083\)
\(L(\frac12)\) \(\approx\) \(1.968964083\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
good2 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
3 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 - T )^{16}( 1 + T )^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 - T )^{16}( 1 + T )^{16} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
53 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
61 \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
89 \( ( 1 - T )^{16}( 1 + T )^{16} \)
97 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
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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

−2.90558334796550583097059412773, −2.84700666374357680077063875985, −2.66271274303901623638327362446, −2.62399262297717839404203554524, −2.53813625316108690803939248694, −2.49537051034919396804455674667, −2.20755853854052486369819765413, −2.17888719356151531130453089708, −2.12913614053807361852304908694, −2.12808408956816070720868881180, −2.11020156993529458127967562879, −2.10352342462384947649298661910, −2.05735447972272226059266677709, −1.66805489594990129007048250600, −1.66268835799979174627563984981, −1.54725942861905567362802865192, −1.52602046161998704417753219774, −1.51047484091680220904798594506, −1.48741450293689389947942293237, −1.14336232502216317532529679802, −1.12502795633067642460362405588, −1.09518782375882725800935842988, −1.08260855800417370324405127814, −0.850581854587085936778647363934, −0.812487694459108776906475400755, 0.812487694459108776906475400755, 0.850581854587085936778647363934, 1.08260855800417370324405127814, 1.09518782375882725800935842988, 1.12502795633067642460362405588, 1.14336232502216317532529679802, 1.48741450293689389947942293237, 1.51047484091680220904798594506, 1.52602046161998704417753219774, 1.54725942861905567362802865192, 1.66268835799979174627563984981, 1.66805489594990129007048250600, 2.05735447972272226059266677709, 2.10352342462384947649298661910, 2.11020156993529458127967562879, 2.12808408956816070720868881180, 2.12913614053807361852304908694, 2.17888719356151531130453089708, 2.20755853854052486369819765413, 2.49537051034919396804455674667, 2.53813625316108690803939248694, 2.62399262297717839404203554524, 2.66271274303901623638327362446, 2.84700666374357680077063875985, 2.90558334796550583097059412773

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

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