Properties

Degree 16
Conductor $ 3^{16} \cdot 5^{8} \cdot 89^{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  − 2-s − 4·4-s + 8·5-s − 6·7-s + 4·8-s − 8·10-s − 14·11-s − 7·13-s + 6·14-s + 7·16-s − 17·17-s + 17·19-s − 32·20-s + 14·22-s + 23-s + 36·25-s + 7·26-s + 24·28-s − 10·29-s + 31-s − 6·32-s + 17·34-s − 48·35-s − 11·37-s − 17·38-s + 32·40-s − 15·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 2·4-s + 3.57·5-s − 2.26·7-s + 1.41·8-s − 2.52·10-s − 4.22·11-s − 1.94·13-s + 1.60·14-s + 7/4·16-s − 4.12·17-s + 3.90·19-s − 7.15·20-s + 2.98·22-s + 0.208·23-s + 36/5·25-s + 1.37·26-s + 4.53·28-s − 1.85·29-s + 0.179·31-s − 1.06·32-s + 2.91·34-s − 8.11·35-s − 1.80·37-s − 2.75·38-s + 5.05·40-s − 2.34·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 89^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 89^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{16} \cdot 5^{8} \cdot 89^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4005} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  8
Selberg data  =  $(16,\ 3^{16} \cdot 5^{8} \cdot 89^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{3,\;5,\;89\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad3 \( 1 \)
5 \( ( 1 - T )^{8} \)
89 \( ( 1 + T )^{8} \)
good2 \( 1 + T + 5 T^{2} + 5 T^{3} + 7 p T^{4} + 13 T^{5} + 33 T^{6} + 23 T^{7} + 67 T^{8} + 23 p T^{9} + 33 p^{2} T^{10} + 13 p^{3} T^{11} + 7 p^{5} T^{12} + 5 p^{5} T^{13} + 5 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 879 T^{4} + 3140 T^{5} + 10940 T^{6} + 32450 T^{7} + 91724 T^{8} + 32450 p T^{9} + 10940 p^{2} T^{10} + 3140 p^{3} T^{11} + 879 p^{4} T^{12} + 192 p^{5} T^{13} + 44 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 14 T + 138 T^{2} + 1003 T^{3} + 6089 T^{4} + 31337 T^{5} + 141122 T^{6} + 50768 p T^{7} + 1966468 T^{8} + 50768 p^{2} T^{9} + 141122 p^{2} T^{10} + 31337 p^{3} T^{11} + 6089 p^{4} T^{12} + 1003 p^{5} T^{13} + 138 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 7 T + 64 T^{2} + 367 T^{3} + 2087 T^{4} + 9437 T^{5} + 42924 T^{6} + 166605 T^{7} + 634648 T^{8} + 166605 p T^{9} + 42924 p^{2} T^{10} + 9437 p^{3} T^{11} + 2087 p^{4} T^{12} + 367 p^{5} T^{13} + 64 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + p T + 201 T^{2} + 1786 T^{3} + 13337 T^{4} + 84604 T^{5} + 470691 T^{6} + 2309209 T^{7} + 594148 p T^{8} + 2309209 p T^{9} + 470691 p^{2} T^{10} + 84604 p^{3} T^{11} + 13337 p^{4} T^{12} + 1786 p^{5} T^{13} + 201 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \)
19 \( 1 - 17 T + 232 T^{2} - 2093 T^{3} + 16535 T^{4} - 104185 T^{5} + 602904 T^{6} - 2978949 T^{7} + 13901644 T^{8} - 2978949 p T^{9} + 602904 p^{2} T^{10} - 104185 p^{3} T^{11} + 16535 p^{4} T^{12} - 2093 p^{5} T^{13} + 232 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - T + 117 T^{2} - 190 T^{3} + 6873 T^{4} - 540 p T^{5} + 264599 T^{6} - 449267 T^{7} + 7188320 T^{8} - 449267 p T^{9} + 264599 p^{2} T^{10} - 540 p^{4} T^{11} + 6873 p^{4} T^{12} - 190 p^{5} T^{13} + 117 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 10 T + 134 T^{2} + 1121 T^{3} + 9999 T^{4} + 67781 T^{5} + 487566 T^{6} + 2792852 T^{7} + 16567336 T^{8} + 2792852 p T^{9} + 487566 p^{2} T^{10} + 67781 p^{3} T^{11} + 9999 p^{4} T^{12} + 1121 p^{5} T^{13} + 134 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - T + 112 T^{2} - 377 T^{3} + 6885 T^{4} - 30457 T^{5} + 330560 T^{6} - 1332593 T^{7} + 12027896 T^{8} - 1332593 p T^{9} + 330560 p^{2} T^{10} - 30457 p^{3} T^{11} + 6885 p^{4} T^{12} - 377 p^{5} T^{13} + 112 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 11 T + 271 T^{2} + 2296 T^{3} + 31701 T^{4} + 215486 T^{5} + 2164581 T^{6} + 12102083 T^{7} + 97087004 T^{8} + 12102083 p T^{9} + 2164581 p^{2} T^{10} + 215486 p^{3} T^{11} + 31701 p^{4} T^{12} + 2296 p^{5} T^{13} + 271 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 15 T + 323 T^{2} + 3678 T^{3} + 46003 T^{4} + 408932 T^{5} + 3748585 T^{6} + 26615591 T^{7} + 191604672 T^{8} + 26615591 p T^{9} + 3748585 p^{2} T^{10} + 408932 p^{3} T^{11} + 46003 p^{4} T^{12} + 3678 p^{5} T^{13} + 323 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 5 T + 148 T^{2} + 727 T^{3} + 11213 T^{4} + 70591 T^{5} + 635332 T^{6} + 4550341 T^{7} + 29224856 T^{8} + 4550341 p T^{9} + 635332 p^{2} T^{10} + 70591 p^{3} T^{11} + 11213 p^{4} T^{12} + 727 p^{5} T^{13} + 148 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 12 T + 296 T^{2} + 2797 T^{3} + 41203 T^{4} + 319943 T^{5} + 3501620 T^{6} + 22665974 T^{7} + 199070992 T^{8} + 22665974 p T^{9} + 3501620 p^{2} T^{10} + 319943 p^{3} T^{11} + 41203 p^{4} T^{12} + 2797 p^{5} T^{13} + 296 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - T + 138 T^{2} - 13 T^{3} + 13219 T^{4} - 2079 T^{5} + 972322 T^{6} + 230941 T^{7} + 55597856 T^{8} + 230941 p T^{9} + 972322 p^{2} T^{10} - 2079 p^{3} T^{11} + 13219 p^{4} T^{12} - 13 p^{5} T^{13} + 138 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 26 T + 668 T^{2} + 10544 T^{3} + 156775 T^{4} + 1786628 T^{5} + 19212548 T^{6} + 169919486 T^{7} + 1425687308 T^{8} + 169919486 p T^{9} + 19212548 p^{2} T^{10} + 1786628 p^{3} T^{11} + 156775 p^{4} T^{12} + 10544 p^{5} T^{13} + 668 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 13 T + 237 T^{2} - 2462 T^{3} + 34631 T^{4} - 298904 T^{5} + 3229159 T^{6} - 24630041 T^{7} + 232136776 T^{8} - 24630041 p T^{9} + 3229159 p^{2} T^{10} - 298904 p^{3} T^{11} + 34631 p^{4} T^{12} - 2462 p^{5} T^{13} + 237 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 25 T + 349 T^{2} + 3294 T^{3} + 26345 T^{4} + 201780 T^{5} + 1492455 T^{6} + 10054003 T^{7} + 74148916 T^{8} + 10054003 p T^{9} + 1492455 p^{2} T^{10} + 201780 p^{3} T^{11} + 26345 p^{4} T^{12} + 3294 p^{5} T^{13} + 349 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 28 T + 854 T^{2} + 15168 T^{3} + 264553 T^{4} + 3402968 T^{5} + 42092278 T^{6} + 412645268 T^{7} + 3870356788 T^{8} + 412645268 p T^{9} + 42092278 p^{2} T^{10} + 3402968 p^{3} T^{11} + 264553 p^{4} T^{12} + 15168 p^{5} T^{13} + 854 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 17 T + 445 T^{2} + 5624 T^{3} + 89597 T^{4} + 905898 T^{5} + 11091175 T^{6} + 94140709 T^{7} + 954364028 T^{8} + 94140709 p T^{9} + 11091175 p^{2} T^{10} + 905898 p^{3} T^{11} + 89597 p^{4} T^{12} + 5624 p^{5} T^{13} + 445 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 7 T + 361 T^{2} + 1674 T^{3} + 61897 T^{4} + 219580 T^{5} + 7407291 T^{6} + 23371865 T^{7} + 676053924 T^{8} + 23371865 p T^{9} + 7407291 p^{2} T^{10} + 219580 p^{3} T^{11} + 61897 p^{4} T^{12} + 1674 p^{5} T^{13} + 361 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 44 T + 1283 T^{2} + 26198 T^{3} + 433257 T^{4} + 5890402 T^{5} + 69986761 T^{6} + 732699384 T^{7} + 7013950512 T^{8} + 732699384 p T^{9} + 69986761 p^{2} T^{10} + 5890402 p^{3} T^{11} + 433257 p^{4} T^{12} + 26198 p^{5} T^{13} + 1283 p^{6} T^{14} + 44 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - T + 420 T^{2} - 1057 T^{3} + 88659 T^{4} - 308543 T^{5} + 13073720 T^{6} - 46945695 T^{7} + 1458833424 T^{8} - 46945695 p T^{9} + 13073720 p^{2} T^{10} - 308543 p^{3} T^{11} + 88659 p^{4} T^{12} - 1057 p^{5} T^{13} + 420 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.15179138127968518965138624950, −3.57228933279433656146416262399, −3.49985222781124143855024423158, −3.48495091435878125806561972854, −3.30525674727906986117502877489, −3.30238711904642034757048576045, −3.17007105188439227936852841796, −3.10989391739364177464801798496, −2.91239416510549673016004835419, −2.77588875433909425430119883925, −2.76046409449729015729018324750, −2.72829893867766070638604148700, −2.68011431311991739023121081637, −2.40029136759666425083287819783, −2.29828788684887928615226345584, −2.29161829484172047694955294984, −2.13208635336916323987514522934, −1.91200569855489123789239403615, −1.70465997737705327945372653031, −1.52985507192588514682833156098, −1.43258806366851688126908271421, −1.42994015789105176831538915534, −1.26841680487887209092901813962, −1.18246191249945642528772407619, −1.09918351396455551656916745411, 0, 0, 0, 0, 0, 0, 0, 0, 1.09918351396455551656916745411, 1.18246191249945642528772407619, 1.26841680487887209092901813962, 1.42994015789105176831538915534, 1.43258806366851688126908271421, 1.52985507192588514682833156098, 1.70465997737705327945372653031, 1.91200569855489123789239403615, 2.13208635336916323987514522934, 2.29161829484172047694955294984, 2.29828788684887928615226345584, 2.40029136759666425083287819783, 2.68011431311991739023121081637, 2.72829893867766070638604148700, 2.76046409449729015729018324750, 2.77588875433909425430119883925, 2.91239416510549673016004835419, 3.10989391739364177464801798496, 3.17007105188439227936852841796, 3.30238711904642034757048576045, 3.30525674727906986117502877489, 3.48495091435878125806561972854, 3.49985222781124143855024423158, 3.57228933279433656146416262399, 4.15179138127968518965138624950

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

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