Properties

Degree 16
Conductor $ 3^{8} \cdot 7^{16} \cdot 41^{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  + 2-s + 8·3-s − 4-s + 7·5-s + 8·6-s + 36·9-s + 7·10-s + 11·11-s − 8·12-s + 10·13-s + 56·15-s − 9·16-s + 3·17-s + 36·18-s + 6·19-s − 7·20-s + 11·22-s + 14·23-s + 17·25-s + 10·26-s + 120·27-s + 2·29-s + 56·30-s + 16·31-s − 13·32-s + 88·33-s + 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 4.61·3-s − 1/2·4-s + 3.13·5-s + 3.26·6-s + 12·9-s + 2.21·10-s + 3.31·11-s − 2.30·12-s + 2.77·13-s + 14.4·15-s − 9/4·16-s + 0.727·17-s + 8.48·18-s + 1.37·19-s − 1.56·20-s + 2.34·22-s + 2.91·23-s + 17/5·25-s + 1.96·26-s + 23.0·27-s + 0.371·29-s + 10.2·30-s + 2.87·31-s − 2.29·32-s + 15.3·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 7^{16} \cdot 41^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 3^{8} \cdot 7^{16} \cdot 41^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $3906.095601$
$L(\frac12)$  $\approx$  $3906.095601$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( ( 1 - T )^{8} \)
7 \( 1 \)
41 \( ( 1 - T )^{8} \)
good2 \( 1 - T + p T^{2} - 3 T^{3} + 7 p T^{4} - 13 T^{5} + 5 p^{2} T^{6} - 15 p T^{7} + 81 T^{8} - 15 p^{2} T^{9} + 5 p^{4} T^{10} - 13 p^{3} T^{11} + 7 p^{5} T^{12} - 3 p^{5} T^{13} + p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
5 \( 1 - 7 T + 32 T^{2} - 104 T^{3} + 291 T^{4} - 739 T^{5} + 1773 T^{6} - 4004 T^{7} + 8963 T^{8} - 4004 p T^{9} + 1773 p^{2} T^{10} - 739 p^{3} T^{11} + 291 p^{4} T^{12} - 104 p^{5} T^{13} + 32 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - p T + 109 T^{2} - 722 T^{3} + 4327 T^{4} - 21132 T^{5} + 94519 T^{6} - 364514 T^{7} + 1293423 T^{8} - 364514 p T^{9} + 94519 p^{2} T^{10} - 21132 p^{3} T^{11} + 4327 p^{4} T^{12} - 722 p^{5} T^{13} + 109 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \)
13 \( ( 1 - 5 T + 42 T^{2} - 151 T^{3} + 803 T^{4} - 151 p T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 3 T + 88 T^{2} - 249 T^{3} + 3628 T^{4} - 10429 T^{5} + 97377 T^{6} - 16168 p T^{7} + 1911047 T^{8} - 16168 p^{2} T^{9} + 97377 p^{2} T^{10} - 10429 p^{3} T^{11} + 3628 p^{4} T^{12} - 249 p^{5} T^{13} + 88 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 6 T + 67 T^{2} - 389 T^{3} + 2468 T^{4} - 11408 T^{5} + 61405 T^{6} - 240975 T^{7} + 1214249 T^{8} - 240975 p T^{9} + 61405 p^{2} T^{10} - 11408 p^{3} T^{11} + 2468 p^{4} T^{12} - 389 p^{5} T^{13} + 67 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 14 T + 188 T^{2} - 1660 T^{3} + 13785 T^{4} - 92987 T^{5} + 590823 T^{6} - 3228139 T^{7} + 16594745 T^{8} - 3228139 p T^{9} + 590823 p^{2} T^{10} - 92987 p^{3} T^{11} + 13785 p^{4} T^{12} - 1660 p^{5} T^{13} + 188 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 2 T + 86 T^{2} - 153 T^{3} + 3409 T^{4} - 670 T^{5} + 83827 T^{6} + 239476 T^{7} + 1987143 T^{8} + 239476 p T^{9} + 83827 p^{2} T^{10} - 670 p^{3} T^{11} + 3409 p^{4} T^{12} - 153 p^{5} T^{13} + 86 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 16 T + 266 T^{2} - 2949 T^{3} + 29580 T^{4} - 246472 T^{5} + 59579 p T^{6} - 12083235 T^{7} + 71657915 T^{8} - 12083235 p T^{9} + 59579 p^{3} T^{10} - 246472 p^{3} T^{11} + 29580 p^{4} T^{12} - 2949 p^{5} T^{13} + 266 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 20 T + 325 T^{2} + 3627 T^{3} + 34697 T^{4} + 274640 T^{5} + 1965177 T^{6} + 12776670 T^{7} + 79575859 T^{8} + 12776670 p T^{9} + 1965177 p^{2} T^{10} + 274640 p^{3} T^{11} + 34697 p^{4} T^{12} + 3627 p^{5} T^{13} + 325 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 7 T + 176 T^{2} - 1539 T^{3} + 17898 T^{4} - 148602 T^{5} + 1313572 T^{6} - 8856203 T^{7} + 1587107 p T^{8} - 8856203 p T^{9} + 1313572 p^{2} T^{10} - 148602 p^{3} T^{11} + 17898 p^{4} T^{12} - 1539 p^{5} T^{13} + 176 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 14 T + 344 T^{2} - 3788 T^{3} + 52751 T^{4} - 468163 T^{5} + 4734300 T^{6} - 34338524 T^{7} + 273639553 T^{8} - 34338524 p T^{9} + 4734300 p^{2} T^{10} - 468163 p^{3} T^{11} + 52751 p^{4} T^{12} - 3788 p^{5} T^{13} + 344 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 7 T + 329 T^{2} - 2227 T^{3} + 52589 T^{4} - 315438 T^{5} + 5144510 T^{6} - 26357108 T^{7} + 331902573 T^{8} - 26357108 p T^{9} + 5144510 p^{2} T^{10} - 315438 p^{3} T^{11} + 52589 p^{4} T^{12} - 2227 p^{5} T^{13} + 329 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 22 T + 451 T^{2} - 5987 T^{3} + 76223 T^{4} - 765180 T^{5} + 7510827 T^{6} - 62437904 T^{7} + 516169223 T^{8} - 62437904 p T^{9} + 7510827 p^{2} T^{10} - 765180 p^{3} T^{11} + 76223 p^{4} T^{12} - 5987 p^{5} T^{13} + 451 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 235 T^{2} - 219 T^{3} + 22651 T^{4} - 72714 T^{5} + 1207263 T^{6} - 9201786 T^{7} + 57784175 T^{8} - 9201786 p T^{9} + 1207263 p^{2} T^{10} - 72714 p^{3} T^{11} + 22651 p^{4} T^{12} - 219 p^{5} T^{13} + 235 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 12 T + 382 T^{2} - 3513 T^{3} + 69384 T^{4} - 542920 T^{5} + 8067795 T^{6} - 53363601 T^{7} + 642294989 T^{8} - 53363601 p T^{9} + 8067795 p^{2} T^{10} - 542920 p^{3} T^{11} + 69384 p^{4} T^{12} - 3513 p^{5} T^{13} + 382 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 5 T + 470 T^{2} + 2248 T^{3} + 101408 T^{4} + 446526 T^{5} + 13192902 T^{6} + 50973626 T^{7} + 1136623295 T^{8} + 50973626 p T^{9} + 13192902 p^{2} T^{10} + 446526 p^{3} T^{11} + 101408 p^{4} T^{12} + 2248 p^{5} T^{13} + 470 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 2 T + 327 T^{2} + 241 T^{3} + 51485 T^{4} + 107105 T^{5} + 5880012 T^{6} + 11437223 T^{7} + 507231001 T^{8} + 11437223 p T^{9} + 5880012 p^{2} T^{10} + 107105 p^{3} T^{11} + 51485 p^{4} T^{12} + 241 p^{5} T^{13} + 327 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 15 T + 567 T^{2} + 6821 T^{3} + 142715 T^{4} + 1421020 T^{5} + 21217240 T^{6} + 175433474 T^{7} + 2051476003 T^{8} + 175433474 p T^{9} + 21217240 p^{2} T^{10} + 1421020 p^{3} T^{11} + 142715 p^{4} T^{12} + 6821 p^{5} T^{13} + 567 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 15 T + 621 T^{2} - 7591 T^{3} + 170269 T^{4} - 1716666 T^{5} + 27194354 T^{6} - 225973994 T^{7} + 2783475951 T^{8} - 225973994 p T^{9} + 27194354 p^{2} T^{10} - 1716666 p^{3} T^{11} + 170269 p^{4} T^{12} - 7591 p^{5} T^{13} + 621 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 29 T + 745 T^{2} - 12702 T^{3} + 197115 T^{4} - 2469540 T^{5} + 29311002 T^{6} - 301308421 T^{7} + 3020587875 T^{8} - 301308421 p T^{9} + 29311002 p^{2} T^{10} - 2469540 p^{3} T^{11} + 197115 p^{4} T^{12} - 12702 p^{5} T^{13} + 745 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 19 T + 746 T^{2} - 10988 T^{3} + 243141 T^{4} - 2884285 T^{5} + 45851095 T^{6} - 443206010 T^{7} + 5504083913 T^{8} - 443206010 p T^{9} + 45851095 p^{2} T^{10} - 2884285 p^{3} T^{11} + 243141 p^{4} T^{12} - 10988 p^{5} T^{13} + 746 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.24764013580982813222844224424, −3.15065839005923606214549796080, −2.99790518799137699470984861850, −2.95487372908940787764055850383, −2.87126707878407130751757385625, −2.86479688708274493416816186122, −2.47844486287393200055789579772, −2.45526706181324065171226750647, −2.44328963937335634681052745926, −2.32831049031147362564389318057, −2.26402994211093032574895750899, −2.08786049976310395982685198149, −1.94076791486321292222352094319, −1.77832969501879149481219902759, −1.72967010409129227262335195479, −1.70854597789136532611494387765, −1.51134496415935205468792478518, −1.38766469162176339825603674524, −1.35347719470437705792573388618, −1.00741684846325318275026327863, −0.903489459514833841238851719565, −0.891506266224048905905734494936, −0.862959899299200933696688549984, −0.70645246722566756506017658707, −0.48973628615642097886555838330, 0.48973628615642097886555838330, 0.70645246722566756506017658707, 0.862959899299200933696688549984, 0.891506266224048905905734494936, 0.903489459514833841238851719565, 1.00741684846325318275026327863, 1.35347719470437705792573388618, 1.38766469162176339825603674524, 1.51134496415935205468792478518, 1.70854597789136532611494387765, 1.72967010409129227262335195479, 1.77832969501879149481219902759, 1.94076791486321292222352094319, 2.08786049976310395982685198149, 2.26402994211093032574895750899, 2.32831049031147362564389318057, 2.44328963937335634681052745926, 2.45526706181324065171226750647, 2.47844486287393200055789579772, 2.86479688708274493416816186122, 2.87126707878407130751757385625, 2.95487372908940787764055850383, 2.99790518799137699470984861850, 3.15065839005923606214549796080, 3.24764013580982813222844224424

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

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