Properties

Degree 16
Conductor $ 2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 8

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 7·3-s + 36·4-s + 8·5-s + 56·6-s − 6·7-s − 120·8-s + 16·9-s − 64·10-s + 8·11-s − 252·12-s − 2·13-s + 48·14-s − 56·15-s + 330·16-s − 8·17-s − 128·18-s + 288·20-s + 42·21-s − 64·22-s − 18·23-s + 840·24-s + 36·25-s + 16·26-s + 2·27-s − 216·28-s + 8·29-s + ⋯
L(s)  = 1  − 5.65·2-s − 4.04·3-s + 18·4-s + 3.57·5-s + 22.8·6-s − 2.26·7-s − 42.4·8-s + 16/3·9-s − 20.2·10-s + 2.41·11-s − 72.7·12-s − 0.554·13-s + 12.8·14-s − 14.4·15-s + 82.5·16-s − 1.94·17-s − 30.1·18-s + 64.3·20-s + 9.16·21-s − 13.6·22-s − 3.75·23-s + 171.·24-s + 36/5·25-s + 3.13·26-s + 0.384·27-s − 40.8·28-s + 1.48·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4730} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  8
Selberg data  =  $(16,\ 2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{8} \)
5 \( ( 1 - T )^{8} \)
11 \( ( 1 - T )^{8} \)
43 \( ( 1 + T )^{8} \)
good3 \( 1 + 7 T + 11 p T^{2} + 13 p^{2} T^{3} + 115 p T^{4} + 98 p^{2} T^{5} + 1996 T^{6} + 4052 T^{7} + 7394 T^{8} + 4052 p T^{9} + 1996 p^{2} T^{10} + 98 p^{5} T^{11} + 115 p^{5} T^{12} + 13 p^{7} T^{13} + 11 p^{7} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 + 6 T + 47 T^{2} + 201 T^{3} + 134 p T^{4} + 3162 T^{5} + 11103 T^{6} + 31303 T^{7} + 91046 T^{8} + 31303 p T^{9} + 11103 p^{2} T^{10} + 3162 p^{3} T^{11} + 134 p^{5} T^{12} + 201 p^{5} T^{13} + 47 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 2 T + 62 T^{2} + 92 T^{3} + 1736 T^{4} + 1868 T^{5} + 30866 T^{6} + 1942 p T^{7} + 430990 T^{8} + 1942 p^{2} T^{9} + 30866 p^{2} T^{10} + 1868 p^{3} T^{11} + 1736 p^{4} T^{12} + 92 p^{5} T^{13} + 62 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 8 T + 117 T^{2} + 781 T^{3} + 6252 T^{4} + 34310 T^{5} + 199135 T^{6} + 896479 T^{7} + 4139526 T^{8} + 896479 p T^{9} + 199135 p^{2} T^{10} + 34310 p^{3} T^{11} + 6252 p^{4} T^{12} + 781 p^{5} T^{13} + 117 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 69 T^{2} - 17 T^{3} + 2296 T^{4} + 54 T^{5} + 50801 T^{6} + 17155 T^{7} + 953410 T^{8} + 17155 p T^{9} + 50801 p^{2} T^{10} + 54 p^{3} T^{11} + 2296 p^{4} T^{12} - 17 p^{5} T^{13} + 69 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 + 18 T + 282 T^{2} + 2920 T^{3} + 26876 T^{4} + 197944 T^{5} + 1320702 T^{6} + 7469186 T^{7} + 38622038 T^{8} + 7469186 p T^{9} + 1320702 p^{2} T^{10} + 197944 p^{3} T^{11} + 26876 p^{4} T^{12} + 2920 p^{5} T^{13} + 282 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 8 T + 98 T^{2} - 882 T^{3} + 7209 T^{4} - 48058 T^{5} + 342856 T^{6} - 2001388 T^{7} + 11097800 T^{8} - 2001388 p T^{9} + 342856 p^{2} T^{10} - 48058 p^{3} T^{11} + 7209 p^{4} T^{12} - 882 p^{5} T^{13} + 98 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 11 T + 223 T^{2} + 1996 T^{3} + 22686 T^{4} + 165386 T^{5} + 1356577 T^{6} + 8071481 T^{7} + 51961410 T^{8} + 8071481 p T^{9} + 1356577 p^{2} T^{10} + 165386 p^{3} T^{11} + 22686 p^{4} T^{12} + 1996 p^{5} T^{13} + 223 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 17 T + 307 T^{2} + 3294 T^{3} + 36336 T^{4} + 298428 T^{5} + 2493261 T^{6} + 16600665 T^{7} + 112085390 T^{8} + 16600665 p T^{9} + 2493261 p^{2} T^{10} + 298428 p^{3} T^{11} + 36336 p^{4} T^{12} + 3294 p^{5} T^{13} + 307 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 12 T + 244 T^{2} - 58 p T^{3} + 28932 T^{4} - 230138 T^{5} + 2108972 T^{6} - 14013792 T^{7} + 103822454 T^{8} - 14013792 p T^{9} + 2108972 p^{2} T^{10} - 230138 p^{3} T^{11} + 28932 p^{4} T^{12} - 58 p^{6} T^{13} + 244 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 19 T + 237 T^{2} + 1565 T^{3} + 9168 T^{4} + 27117 T^{5} + 86891 T^{6} - 1482653 T^{7} - 10377266 T^{8} - 1482653 p T^{9} + 86891 p^{2} T^{10} + 27117 p^{3} T^{11} + 9168 p^{4} T^{12} + 1565 p^{5} T^{13} + 237 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 7 T + 303 T^{2} + 1683 T^{3} + 42385 T^{4} + 194180 T^{5} + 3728662 T^{6} + 14456650 T^{7} + 231474938 T^{8} + 14456650 p T^{9} + 3728662 p^{2} T^{10} + 194180 p^{3} T^{11} + 42385 p^{4} T^{12} + 1683 p^{5} T^{13} + 303 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - T + 297 T^{2} - 83 T^{3} + 41548 T^{4} + 8647 T^{5} + 3752399 T^{6} + 1642201 T^{7} + 251231334 T^{8} + 1642201 p T^{9} + 3752399 p^{2} T^{10} + 8647 p^{3} T^{11} + 41548 p^{4} T^{12} - 83 p^{5} T^{13} + 297 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 6 T + 316 T^{2} - 758 T^{3} + 38007 T^{4} + 66450 T^{5} + 2364782 T^{6} + 16724690 T^{7} + 1990388 p T^{8} + 16724690 p T^{9} + 2364782 p^{2} T^{10} + 66450 p^{3} T^{11} + 38007 p^{4} T^{12} - 758 p^{5} T^{13} + 316 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 22 T + 446 T^{2} + 5574 T^{3} + 67696 T^{4} + 622446 T^{5} + 6081474 T^{6} + 48840078 T^{7} + 436862750 T^{8} + 48840078 p T^{9} + 6081474 p^{2} T^{10} + 622446 p^{3} T^{11} + 67696 p^{4} T^{12} + 5574 p^{5} T^{13} + 446 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 14 T + 287 T^{2} + 3393 T^{3} + 49762 T^{4} + 501326 T^{5} + 5711765 T^{6} + 49212093 T^{7} + 477242794 T^{8} + 49212093 p T^{9} + 5711765 p^{2} T^{10} + 501326 p^{3} T^{11} + 49762 p^{4} T^{12} + 3393 p^{5} T^{13} + 287 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 13 T + 514 T^{2} + 5567 T^{3} + 119936 T^{4} + 1079789 T^{5} + 16553022 T^{6} + 123546039 T^{7} + 1481601022 T^{8} + 123546039 p T^{9} + 16553022 p^{2} T^{10} + 1079789 p^{3} T^{11} + 119936 p^{4} T^{12} + 5567 p^{5} T^{13} + 514 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 8 T + 424 T^{2} + 2987 T^{3} + 91468 T^{4} + 564333 T^{5} + 12518738 T^{6} + 66589874 T^{7} + 1180934250 T^{8} + 66589874 p T^{9} + 12518738 p^{2} T^{10} + 564333 p^{3} T^{11} + 91468 p^{4} T^{12} + 2987 p^{5} T^{13} + 424 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 4 T + 452 T^{2} + 1499 T^{3} + 98766 T^{4} + 288133 T^{5} + 13829002 T^{6} + 35768690 T^{7} + 1357063142 T^{8} + 35768690 p T^{9} + 13829002 p^{2} T^{10} + 288133 p^{3} T^{11} + 98766 p^{4} T^{12} + 1499 p^{5} T^{13} + 452 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 5 T + 299 T^{2} - 508 T^{3} + 49214 T^{4} + 31858 T^{5} + 5570917 T^{6} + 11761287 T^{7} + 540583874 T^{8} + 11761287 p T^{9} + 5570917 p^{2} T^{10} + 31858 p^{3} T^{11} + 49214 p^{4} T^{12} - 508 p^{5} T^{13} + 299 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 23 T + 574 T^{2} + 7585 T^{3} + 105528 T^{4} + 830323 T^{5} + 7761410 T^{6} + 29168789 T^{7} + 391255310 T^{8} + 29168789 p T^{9} + 7761410 p^{2} T^{10} + 830323 p^{3} T^{11} + 105528 p^{4} T^{12} + 7585 p^{5} T^{13} + 574 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.88496911731922832737247520151, −3.43120659158157333485785253056, −3.21604207676774996734985454210, −3.20854368031342544361032400673, −3.19966182872137490508014728116, −3.16078258611842075473906114672, −3.15304496131833790655900421742, −3.03836145619920466371699854399, −3.01705212332249184109349189232, −2.57486162456305705307123485431, −2.42421035640832977565741737549, −2.28302236346254022619138946966, −2.13838263193937889778502939675, −2.13535643350076355741482496870, −2.09417507691815021202547598700, −2.04591937735311447931992583862, −1.94640953834633423784926627253, −1.71580414946402963754479911383, −1.61751233852517058776895457527, −1.33346366859150099764648378213, −1.19532708609762698666342566592, −1.19161153101949589856171613130, −1.08125724014400599212231258384, −1.01342194243731750262531311886, −0.978337269785765959093172738511, 0, 0, 0, 0, 0, 0, 0, 0, 0.978337269785765959093172738511, 1.01342194243731750262531311886, 1.08125724014400599212231258384, 1.19161153101949589856171613130, 1.19532708609762698666342566592, 1.33346366859150099764648378213, 1.61751233852517058776895457527, 1.71580414946402963754479911383, 1.94640953834633423784926627253, 2.04591937735311447931992583862, 2.09417507691815021202547598700, 2.13535643350076355741482496870, 2.13838263193937889778502939675, 2.28302236346254022619138946966, 2.42421035640832977565741737549, 2.57486162456305705307123485431, 3.01705212332249184109349189232, 3.03836145619920466371699854399, 3.15304496131833790655900421742, 3.16078258611842075473906114672, 3.19966182872137490508014728116, 3.20854368031342544361032400673, 3.21604207676774996734985454210, 3.43120659158157333485785253056, 3.88496911731922832737247520151

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

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