Properties

Degree 16
Conductor $ 2^{24} \cdot 3^{24} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 2·7-s + 2·11-s − 8·13-s − 4·17-s − 6·19-s − 2·23-s + 20·25-s − 16·29-s − 6·31-s − 8·35-s + 16·41-s + 20·47-s − 49-s + 10·53-s + 8·55-s − 22·59-s + 2·61-s − 32·65-s + 2·67-s − 44·71-s − 10·73-s − 4·77-s + 8·79-s + 40·83-s − 16·85-s + 16·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s + 0.603·11-s − 2.21·13-s − 0.970·17-s − 1.37·19-s − 0.417·23-s + 4·25-s − 2.97·29-s − 1.07·31-s − 1.35·35-s + 2.49·41-s + 2.91·47-s − 1/7·49-s + 1.37·53-s + 1.07·55-s − 2.86·59-s + 0.256·61-s − 3.96·65-s + 0.244·67-s − 5.22·71-s − 1.17·73-s − 0.455·77-s + 0.900·79-s + 4.39·83-s − 1.73·85-s + 1.69·89-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 3^{24} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1512} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{24} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.214452653\)
\(L(\frac12)\)  \(\approx\)  \(1.214452653\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
7 \( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( ( 1 - 2 T - 4 T^{2} + 4 T^{3} + 19 T^{4} + 4 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 - 2 T - 14 T^{2} + 48 T^{3} - 58 T^{4} - 26 T^{5} - 168 T^{6} - 2902 T^{7} + 26203 T^{8} - 2902 p T^{9} - 168 p^{2} T^{10} - 26 p^{3} T^{11} - 58 p^{4} T^{12} + 48 p^{5} T^{13} - 14 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
13 \( ( 1 + 4 T + 17 T^{2} - 48 T^{3} - 160 T^{4} - 48 p T^{5} + 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 2 T - 4 T^{2} - 52 T^{3} - 293 T^{4} - 52 p T^{5} - 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + 877 T^{4} - 936 T^{5} + 9706 T^{6} + 54060 T^{7} - 159614 T^{8} + 54060 p T^{9} + 9706 p^{2} T^{10} - 936 p^{3} T^{11} + 877 p^{4} T^{12} + 6 p^{6} T^{13} - 5 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 2 T - 10 T^{2} - 72 T^{3} - 922 T^{4} - 2222 T^{5} + 760 T^{6} + 57846 T^{7} + 692635 T^{8} + 57846 p T^{9} + 760 p^{2} T^{10} - 2222 p^{3} T^{11} - 922 p^{4} T^{12} - 72 p^{5} T^{13} - 10 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 8 T + 36 T^{2} + 24 T^{3} - 182 T^{4} + 24 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 6 T - 38 T^{2} - 348 T^{3} + 517 T^{4} + 10584 T^{5} + 34246 T^{6} - 188550 T^{7} - 2230412 T^{8} - 188550 p T^{9} + 34246 p^{2} T^{10} + 10584 p^{3} T^{11} + 517 p^{4} T^{12} - 348 p^{5} T^{13} - 38 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 61 T^{2} - 648 T^{3} + 2017 T^{4} + 31752 T^{5} + 168050 T^{6} - 910440 T^{7} - 7948706 T^{8} - 910440 p T^{9} + 168050 p^{2} T^{10} + 31752 p^{3} T^{11} + 2017 p^{4} T^{12} - 648 p^{5} T^{13} - 61 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 8 T + 60 T^{2} - 456 T^{3} + 2554 T^{4} - 456 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 14 T^{2} - 72 T^{3} - 1881 T^{4} - 72 p T^{5} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 - 20 T + 196 T^{2} - 1464 T^{3} + 7286 T^{4} - 1508 T^{5} - 284064 T^{6} + 3306668 T^{7} - 27412733 T^{8} + 3306668 p T^{9} - 284064 p^{2} T^{10} - 1508 p^{3} T^{11} + 7286 p^{4} T^{12} - 1464 p^{5} T^{13} + 196 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 10 T - 34 T^{2} + 720 T^{3} - 538 T^{4} - 28730 T^{5} + 169816 T^{6} + 937470 T^{7} - 17639549 T^{8} + 937470 p T^{9} + 169816 p^{2} T^{10} - 28730 p^{3} T^{11} - 538 p^{4} T^{12} + 720 p^{5} T^{13} - 34 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 22 T + 106 T^{2} + 48 T^{3} + 14486 T^{4} + 158758 T^{5} + 199560 T^{6} + 3891698 T^{7} + 83505403 T^{8} + 3891698 p T^{9} + 199560 p^{2} T^{10} + 158758 p^{3} T^{11} + 14486 p^{4} T^{12} + 48 p^{5} T^{13} + 106 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 2 T - 90 T^{2} + 116 T^{3} + 4577 T^{4} - 7740 T^{5} + 332338 T^{6} - 11534 T^{7} - 30676068 T^{8} - 11534 p T^{9} + 332338 p^{2} T^{10} - 7740 p^{3} T^{11} + 4577 p^{4} T^{12} + 116 p^{5} T^{13} - 90 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 2 T - 145 T^{2} - 310 T^{3} + 10229 T^{4} + 49232 T^{5} - 447054 T^{6} - 1987140 T^{7} + 26208538 T^{8} - 1987140 p T^{9} - 447054 p^{2} T^{10} + 49232 p^{3} T^{11} + 10229 p^{4} T^{12} - 310 p^{5} T^{13} - 145 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 + 22 T + 402 T^{2} + 4638 T^{3} + 45946 T^{4} + 4638 p T^{5} + 402 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 10 T - 141 T^{2} - 1714 T^{3} + 12677 T^{4} + 151788 T^{5} - 657782 T^{6} - 4715864 T^{7} + 43753554 T^{8} - 4715864 p T^{9} - 657782 p^{2} T^{10} + 151788 p^{3} T^{11} + 12677 p^{4} T^{12} - 1714 p^{5} T^{13} - 141 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 8 T - 201 T^{2} + 920 T^{3} + 28853 T^{4} - 55488 T^{5} - 3083690 T^{6} + 2064928 T^{7} + 256704030 T^{8} + 2064928 p T^{9} - 3083690 p^{2} T^{10} - 55488 p^{3} T^{11} + 28853 p^{4} T^{12} + 920 p^{5} T^{13} - 201 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 - 20 T + 360 T^{2} - 3684 T^{3} + 40318 T^{4} - 3684 p T^{5} + 360 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 16 T - 92 T^{2} + 1248 T^{3} + 22646 T^{4} - 83536 T^{5} - 3013152 T^{6} + 6008080 T^{7} + 235094707 T^{8} + 6008080 p T^{9} - 3013152 p^{2} T^{10} - 83536 p^{3} T^{11} + 22646 p^{4} T^{12} + 1248 p^{5} T^{13} - 92 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( ( 1 + 230 T^{2} + 72 T^{3} + 26415 T^{4} + 72 p T^{5} + 230 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.05865006999489004015271121179, −3.94881070073955876291767163420, −3.87150878006570065046378251233, −3.58420143966260572218206430463, −3.40668487466838139521213483374, −3.34197672292601282561534931894, −3.16064575176413229043287190986, −3.12404453591232983900965592616, −2.96020917831671011774302728021, −2.80316610804178853322130161265, −2.55742229388302588839466958290, −2.52597626993924625247688386114, −2.52410306231776310237805422442, −2.25238859790963106480688488252, −2.13176396265110734717360686173, −1.95522782397663878498349072363, −1.78779798988716656667008847571, −1.71928211349230438129959879187, −1.65227455117624153009864282352, −1.58385892550694854229078788275, −1.01615110955226368189611019865, −0.76507829935207486510080387595, −0.65922131214681156877846534642, −0.61579662651317965462772858530, −0.10230181979852868072474992211, 0.10230181979852868072474992211, 0.61579662651317965462772858530, 0.65922131214681156877846534642, 0.76507829935207486510080387595, 1.01615110955226368189611019865, 1.58385892550694854229078788275, 1.65227455117624153009864282352, 1.71928211349230438129959879187, 1.78779798988716656667008847571, 1.95522782397663878498349072363, 2.13176396265110734717360686173, 2.25238859790963106480688488252, 2.52410306231776310237805422442, 2.52597626993924625247688386114, 2.55742229388302588839466958290, 2.80316610804178853322130161265, 2.96020917831671011774302728021, 3.12404453591232983900965592616, 3.16064575176413229043287190986, 3.34197672292601282561534931894, 3.40668487466838139521213483374, 3.58420143966260572218206430463, 3.87150878006570065046378251233, 3.94881070073955876291767163420, 4.05865006999489004015271121179

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

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