Properties

Degree 16
Conductor $ 2^{24} \cdot 3^{16} \cdot 7^{8} $
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  + 4-s + 8·7-s + 2·8-s − 16-s − 4·17-s − 12·23-s + 8·25-s + 8·28-s + 8·31-s + 8·32-s + 4·41-s + 36·49-s + 16·56-s + 5·64-s − 4·68-s + 28·71-s − 8·73-s − 40·79-s − 20·89-s − 12·92-s + 40·97-s + 8·100-s + 8·103-s − 8·112-s + 8·113-s − 32·119-s + 24·121-s + ⋯
L(s)  = 1  + 1/2·4-s + 3.02·7-s + 0.707·8-s − 1/4·16-s − 0.970·17-s − 2.50·23-s + 8/5·25-s + 1.51·28-s + 1.43·31-s + 1.41·32-s + 0.624·41-s + 36/7·49-s + 2.13·56-s + 5/8·64-s − 0.485·68-s + 3.32·71-s − 0.936·73-s − 4.50·79-s − 2.11·89-s − 1.25·92-s + 4.06·97-s + 4/5·100-s + 0.788·103-s − 0.755·112-s + 0.752·113-s − 2.93·119-s + 2.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 3^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{504} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{24} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $12.1861$
$L(\frac12)$  $\approx$  $12.1861$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 - T^{2} - p T^{3} + p T^{4} - p^{2} T^{5} - p^{2} T^{6} + p^{4} T^{8} \)
3 \( 1 \)
7 \( ( 1 - T )^{8} \)
good5 \( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 168 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 8312 p^{2} T^{10} + 592 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 26704 p^{2} T^{10} + 1340 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 2 T + 38 T^{2} + 70 T^{3} + 706 T^{4} + 70 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 42432 p^{2} T^{10} + 2012 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 334 p T^{5} + 74 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 34928 p^{2} T^{10} + 2876 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 4 T + 80 T^{2} - 244 T^{3} + 3294 T^{4} - 244 p T^{5} + 80 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1175872 p^{2} T^{10} + 18716 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 214 p T^{5} + 134 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1026520 p^{2} T^{10} + 15676 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 128 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 5027376 p^{2} T^{10} + 52604 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \)
61 \( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 3030896 p^{2} T^{10} + 35516 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 6783272 p^{2} T^{10} + 59996 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 1686 p T^{5} + 194 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 580 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 3972 p T^{5} + 340 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 248 T^{2} + 532 p T^{4} - 5204488 T^{6} + 499369126 T^{8} - 5204488 p^{2} T^{10} + 532 p^{5} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 494 p T^{5} + 198 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 2572 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.78080564686460801494978131890, −4.71117536544338130968831507296, −4.42073875787453191274738032493, −4.35997836515755173218434722486, −4.32453492078120073288991482209, −4.27696268260208546521098781286, −4.10103043067736742002546404459, −3.82460985056023563629016957962, −3.78375436680447551412573481508, −3.59059082676602374985579561688, −3.17910039594386657395572919818, −3.08878082193135043807338288817, −3.00232858290797988011737880139, −2.72351971551395425755218601783, −2.63298668947660401063181106926, −2.58118396245048387723952940934, −2.08762099808199299939956451658, −1.98894732046040587315788481101, −1.92146194737866490544514208631, −1.82482043756576074841121446796, −1.54828611847165281916656256033, −1.54322462237772414681169712156, −0.820140150956060062796269478836, −0.75310448963410233146299673247, −0.69810166096243706579541683139, 0.69810166096243706579541683139, 0.75310448963410233146299673247, 0.820140150956060062796269478836, 1.54322462237772414681169712156, 1.54828611847165281916656256033, 1.82482043756576074841121446796, 1.92146194737866490544514208631, 1.98894732046040587315788481101, 2.08762099808199299939956451658, 2.58118396245048387723952940934, 2.63298668947660401063181106926, 2.72351971551395425755218601783, 3.00232858290797988011737880139, 3.08878082193135043807338288817, 3.17910039594386657395572919818, 3.59059082676602374985579561688, 3.78375436680447551412573481508, 3.82460985056023563629016957962, 4.10103043067736742002546404459, 4.27696268260208546521098781286, 4.32453492078120073288991482209, 4.35997836515755173218434722486, 4.42073875787453191274738032493, 4.71117536544338130968831507296, 4.78080564686460801494978131890

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

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