Properties

Degree 16
Conductor $ 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·16-s + 8·31-s + 4·61-s + 4·121-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 + ⋯
L(s)  = 1  + 2·16-s + 8·31-s + 4·61-s + 4·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1575} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.913063835\)
\(L(\frac12)\)  \(\approx\)  \(1.913063835\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T^{4} + T^{8} \)
good2 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.32921940695173903882516054259, −4.01210138426191986275298638833, −3.87278746461644785201095187842, −3.83311240615520857392924160459, −3.57347278219466344129823894733, −3.57228677658415322600917372568, −3.55187887649718642116361517769, −3.46911862137714315463332035950, −3.27825115821248863519506582903, −3.05256555670321683265318787858, −2.88198707298615110866155074521, −2.70076653172242960124420367632, −2.65949402407712535131607269334, −2.46631360104468861330299454859, −2.43098139386542800302091404075, −2.39203838478409052843605799611, −2.26155972493654590900433537393, −2.08500688479143862743811439639, −1.58738803157816210270948662783, −1.57703436320421469760729459561, −1.27172041915631068316538179779, −1.09775044191987694214685054986, −1.03032890668133285385043912399, −0.949418929156009399994329756279, −0.792180116850680924500338586414, 0.792180116850680924500338586414, 0.949418929156009399994329756279, 1.03032890668133285385043912399, 1.09775044191987694214685054986, 1.27172041915631068316538179779, 1.57703436320421469760729459561, 1.58738803157816210270948662783, 2.08500688479143862743811439639, 2.26155972493654590900433537393, 2.39203838478409052843605799611, 2.43098139386542800302091404075, 2.46631360104468861330299454859, 2.65949402407712535131607269334, 2.70076653172242960124420367632, 2.88198707298615110866155074521, 3.05256555670321683265318787858, 3.27825115821248863519506582903, 3.46911862137714315463332035950, 3.55187887649718642116361517769, 3.57228677658415322600917372568, 3.57347278219466344129823894733, 3.83311240615520857392924160459, 3.87278746461644785201095187842, 4.01210138426191986275298638833, 4.32921940695173903882516054259

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

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