Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·11-s + 16-s − 36·29-s + 20·31-s − 12·41-s + 36·49-s + 36·59-s + 16·61-s − 18·81-s − 12·101-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 24·176-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 7.23·11-s + 1/4·16-s − 6.68·29-s + 3.59·31-s − 1.87·41-s + 36/7·49-s + 4.68·59-s + 2.04·61-s − 2·81-s − 1.19·101-s + 24.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 1.80·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{450} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(9.21933\)
\(L(\frac12)\)  \(\approx\)  \(9.21933\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( ( 1 + p^{2} T^{4} )^{2} \)
5 \( 1 \)
good7 \( 1 - 36 T^{2} + 634 T^{4} - 7272 T^{6} + 59571 T^{8} - 7272 p^{2} T^{10} + 634 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
13 \( 1 - 36 T^{2} + 682 T^{4} - 9000 T^{6} + 106947 T^{8} - 9000 p^{2} T^{10} + 682 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 14 T^{2} + 339 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 108 T^{2} + 5818 T^{4} - 208440 T^{6} + 5501811 T^{8} - 208440 p^{2} T^{10} + 5818 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 18 T + 188 T^{2} + 1404 T^{3} + 8259 T^{4} + 1404 p T^{5} + 188 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 20 p T^{5} + 40 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 644 T^{4} - 1762266 T^{8} - 644 p^{4} T^{12} + p^{8} T^{16} \)
41 \( ( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 - 72 T^{2} + 1633 T^{4} + 6840 T^{6} - 214704 T^{8} + 6840 p^{2} T^{10} + 1633 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 + 508 T^{4} - 3205722 T^{8} + 508 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 - 18 T + 137 T^{2} - 1242 T^{3} + 12372 T^{4} - 1242 p T^{5} + 137 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( 1 - 216 T^{2} + 26737 T^{4} - 2415960 T^{6} + 174766032 T^{8} - 2415960 p^{2} T^{10} + 26737 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 116 T^{2} + 12474 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 3503 T^{4} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 58 T^{2} - 2877 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 - 432 T^{2} + 91513 T^{4} - 12659760 T^{6} + 1239875616 T^{8} - 12659760 p^{2} T^{10} + 91513 p^{4} T^{12} - 432 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{4} \)
97 \( 1 + 360 T^{2} + 65929 T^{4} + 8182440 T^{6} + 834546960 T^{8} + 8182440 p^{2} T^{10} + 65929 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.90763478506545822171899918589, −4.79173147572037409556315604795, −4.50766125782078015708643201299, −4.10034308208664472557946152105, −4.07446016023179885560100458073, −4.06345827385389773277076613818, −4.02398942855310624887305751029, −3.95707688526735035341383745261, −3.79358616511846554887178833585, −3.69531747571070594427839968087, −3.67261656887416147381458942905, −3.54924131744862026884585389901, −3.16302568869081221084260960040, −2.88697465306370203521149396570, −2.74602102717999797919058787140, −2.45549830396006036687089945719, −2.34219956549591211269832497673, −2.06422295028506845427950275146, −1.98384940408493597502087271947, −1.62467957371775592098680165423, −1.46937538518598516504100010522, −1.33834612304026098915903614959, −1.05742311178366563993550327673, −0.996222834343404558890805546803, −0.54403632112254412588335189628, 0.54403632112254412588335189628, 0.996222834343404558890805546803, 1.05742311178366563993550327673, 1.33834612304026098915903614959, 1.46937538518598516504100010522, 1.62467957371775592098680165423, 1.98384940408493597502087271947, 2.06422295028506845427950275146, 2.34219956549591211269832497673, 2.45549830396006036687089945719, 2.74602102717999797919058787140, 2.88697465306370203521149396570, 3.16302568869081221084260960040, 3.54924131744862026884585389901, 3.67261656887416147381458942905, 3.69531747571070594427839968087, 3.79358616511846554887178833585, 3.95707688526735035341383745261, 4.02398942855310624887305751029, 4.06345827385389773277076613818, 4.07446016023179885560100458073, 4.10034308208664472557946152105, 4.50766125782078015708643201299, 4.79173147572037409556315604795, 4.90763478506545822171899918589

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

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