Properties

Degree 16
Conductor $ 2^{40} \cdot 5^{24} $
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  − 2·13-s − 2·17-s + 2·37-s + 2·53-s + 2·73-s + 81-s − 10·89-s + 2·97-s + 4·101-s + 8·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·13-s − 2·17-s + 2·37-s + 2·53-s + 2·73-s + 81-s − 10·89-s + 2·97-s + 4·101-s + 8·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 5^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{4000} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.8664634610$
$L(\frac12)$  $\approx$  $0.8664634610$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
53 \( ( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 + T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−3.67468320598989029457680935535, −3.66515482966975001347982213215, −3.36066973990904861418209705415, −3.26671263641733394939534914498, −3.22599090325069779975953804756, −3.12291237010837719250098840176, −3.10644174595191372097002541949, −2.79372034512748355614855077828, −2.74425083124120998568537263717, −2.67601069272635916747390973873, −2.44437199663304278581876360370, −2.23057068338761059646618022487, −2.21338227684567102175597901696, −2.20348215493208934222310816228, −2.14314490662034437229156457093, −2.04503085101582213803535825315, −2.03401547895775351476235100903, −1.63611253974029431205156335384, −1.33089168249077969050786231808, −1.30743665669055765764464300510, −1.12404826424890290748201183915, −1.03929345663696397723061347531, −0.898831286554477018862416071550, −0.48071580671073338346511777072, −0.27872918691562738619872904327, 0.27872918691562738619872904327, 0.48071580671073338346511777072, 0.898831286554477018862416071550, 1.03929345663696397723061347531, 1.12404826424890290748201183915, 1.30743665669055765764464300510, 1.33089168249077969050786231808, 1.63611253974029431205156335384, 2.03401547895775351476235100903, 2.04503085101582213803535825315, 2.14314490662034437229156457093, 2.20348215493208934222310816228, 2.21338227684567102175597901696, 2.23057068338761059646618022487, 2.44437199663304278581876360370, 2.67601069272635916747390973873, 2.74425083124120998568537263717, 2.79372034512748355614855077828, 3.10644174595191372097002541949, 3.12291237010837719250098840176, 3.22599090325069779975953804756, 3.26671263641733394939534914498, 3.36066973990904861418209705415, 3.66515482966975001347982213215, 3.67468320598989029457680935535

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

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