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  − 16-s + 4·31-s − 4·41-s − 4·61-s − 16·71-s − 2·81-s + 16·101-s − 8·121-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 + ⋯
L(s)  = 1  − 16-s + 4·31-s − 4·41-s − 4·61-s − 16·71-s − 2·81-s + 16·101-s − 8·121-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 + ⋯

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  =  \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6102520563\)
\(L(\frac12)\)  \(\approx\)  \(0.6102520563\)
\(L(1)\)   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 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good2 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
17 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
31 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 + T )^{16} \)
73 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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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

−4.28421835543227617220362949836, −4.17709363693825053970113050431, −4.14272414368454757616696711833, −3.88627037559105298383878259479, −3.77266632992031315665667382987, −3.35988578309785708866730364071, −3.32193752729075552423907321975, −3.25406256216449905893234168721, −3.23924089804624685617367967777, −3.18708264890501264989660077222, −2.88977262446882646130614433621, −2.82536948496413565288365602940, −2.74422780804718044332332636580, −2.59423705567609324146549979766, −2.50623452848331492808740366331, −2.40365667404626328749019836380, −1.81791459245214076889631739877, −1.73525798924145876999627885531, −1.72536222333760870651217469146, −1.71710089531194582892474345443, −1.54377635240555344170191146008, −1.31296636736101174825138448459, −1.23203151559337109698795576573, −0.70464436383444608646009294590, −0.39008112314382201815796157187, 0.39008112314382201815796157187, 0.70464436383444608646009294590, 1.23203151559337109698795576573, 1.31296636736101174825138448459, 1.54377635240555344170191146008, 1.71710089531194582892474345443, 1.72536222333760870651217469146, 1.73525798924145876999627885531, 1.81791459245214076889631739877, 2.40365667404626328749019836380, 2.50623452848331492808740366331, 2.59423705567609324146549979766, 2.74422780804718044332332636580, 2.82536948496413565288365602940, 2.88977262446882646130614433621, 3.18708264890501264989660077222, 3.23924089804624685617367967777, 3.25406256216449905893234168721, 3.32193752729075552423907321975, 3.35988578309785708866730364071, 3.77266632992031315665667382987, 3.88627037559105298383878259479, 4.14272414368454757616696711833, 4.17709363693825053970113050431, 4.28421835543227617220362949836

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

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