Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s − 4·4-s + 78·9-s − 4·11-s − 48·12-s + 4·16-s + 84·17-s + 108·19-s + 12·23-s + 10·25-s + 360·27-s + 72·29-s − 132·31-s − 48·33-s − 312·36-s − 96·37-s − 112·43-s + 16·44-s − 24·47-s + 48·48-s + 78·49-s + 1.00e3·51-s + 32·53-s + 1.29e3·57-s + 132·59-s + 96·61-s + 16·64-s + ⋯
L(s)  = 1  + 4·3-s − 4-s + 26/3·9-s − 0.363·11-s − 4·12-s + 1/4·16-s + 4.94·17-s + 5.68·19-s + 0.521·23-s + 2/5·25-s + 40/3·27-s + 2.48·29-s − 4.25·31-s − 1.45·33-s − 8.66·36-s − 2.59·37-s − 2.60·43-s + 4/11·44-s − 0.510·47-s + 48-s + 1.59·49-s + 19.7·51-s + 0.603·53-s + 22.7·57-s + 2.23·59-s + 1.57·61-s + 1/4·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{210} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(51.7240\)
\(L(\frac12)\)  \(\approx\)  \(51.7240\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
3 \( ( 1 - p T + p T^{2} )^{4} \)
5 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7 \( 1 - 78 T^{2} + 107 p^{2} T^{4} - 78 p^{4} T^{6} + p^{8} T^{8} \)
good11 \( 1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 32486668 p^{2} T^{9} + 2353408 p^{4} T^{10} - 341700 p^{6} T^{11} + 16250 p^{8} T^{12} + 1928 p^{10} T^{13} - 168 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 95438800 p^{4} T^{10} + 357498 p^{8} T^{12} - 860 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 - 84 T + 4208 T^{2} - 155904 T^{3} + 4746250 T^{4} - 123749124 T^{5} + 2818194464 T^{6} - 56792172564 T^{7} + 1020121281523 T^{8} - 56792172564 p^{2} T^{9} + 2818194464 p^{4} T^{10} - 123749124 p^{6} T^{11} + 4746250 p^{8} T^{12} - 155904 p^{10} T^{13} + 4208 p^{12} T^{14} - 84 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 100240172052 p^{2} T^{9} + 4660679966 p^{4} T^{10} - 198492408 p^{6} T^{11} + 7548345 p^{8} T^{12} - 243432 p^{10} T^{13} + 6142 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} \)
23 \( 1 - 12 T - 1512 T^{2} + 15144 T^{3} + 1340890 T^{4} - 9662580 T^{5} - 850652928 T^{6} + 2479269564 T^{7} + 454065720819 T^{8} + 2479269564 p^{2} T^{9} - 850652928 p^{4} T^{10} - 9662580 p^{6} T^{11} + 1340890 p^{8} T^{12} + 15144 p^{10} T^{13} - 1512 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \)
29 \( ( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 82956 p^{2} T^{5} + 2904 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2637318955548 p^{2} T^{9} + 72524765390 p^{4} T^{10} + 1778529960 p^{6} T^{11} + 38602473 p^{8} T^{12} + 722040 p^{10} T^{13} + 11278 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16} \)
37 \( 1 + 96 T + 30 p T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 170035213440 p^{2} T^{9} - 3532027338 p^{4} T^{10} + 360085920 p^{6} T^{11} + 7652401 p^{8} T^{12} - 46464 p^{10} T^{13} + 30 p^{13} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \)
41 \( 1 - 5456 T^{2} + 486924 p T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 49821420976 p^{4} T^{10} + 486924 p^{9} T^{12} - 5456 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 180688 p^{2} T^{5} + 5082 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 + 24 T + 3580 T^{2} + 81312 T^{3} + 3430986 T^{4} + 200307384 T^{5} + 115413136 p T^{6} + 942687331128 T^{7} + 25778360177363 T^{8} + 942687331128 p^{2} T^{9} + 115413136 p^{5} T^{10} + 200307384 p^{6} T^{11} + 3430986 p^{8} T^{12} + 81312 p^{10} T^{13} + 3580 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 - 32 T - 3572 T^{2} + 570944 T^{3} - 366070 T^{4} - 2018388320 T^{5} + 123180800432 T^{6} + 3809526158624 T^{7} - 445561676340941 T^{8} + 3809526158624 p^{2} T^{9} + 123180800432 p^{4} T^{10} - 2018388320 p^{6} T^{11} - 366070 p^{8} T^{12} + 570944 p^{10} T^{13} - 3572 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \)
59 \( 1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 7743785317668 p^{2} T^{9} + 218386477856 p^{4} T^{10} - 4629146772 p^{6} T^{11} + 88851370 p^{8} T^{12} - 1296768 p^{10} T^{13} + 15632 p^{12} T^{14} - 132 p^{14} T^{15} + p^{16} T^{16} \)
61 \( 1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 13588740778464 p^{2} T^{9} + 197844384848 p^{4} T^{10} - 3862327392 p^{6} T^{11} + 60840138 p^{8} T^{12} - 797568 p^{10} T^{13} + 11380 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 2848051223400 p^{2} T^{9} - 240439443082 p^{4} T^{10} + 4026690720 p^{6} T^{11} + 46360561 p^{8} T^{12} - 749040 p^{10} T^{13} - 4522 p^{12} T^{14} + 120 p^{14} T^{15} + p^{16} T^{16} \)
71 \( ( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 338828 p^{2} T^{5} + 13176 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 34150744827912 p^{2} T^{9} - 18252715698 p^{4} T^{10} + 2709573216 p^{6} T^{11} + 47274361 p^{8} T^{12} - 226512 p^{10} T^{13} + 9630 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 10581169939908 p^{2} T^{9} + 66125106126 p^{4} T^{10} - 324907032 p^{6} T^{11} - 58917335 p^{8} T^{12} + 386472 p^{10} T^{13} - 2418 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \)
83 \( 1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 170213772976 p^{4} T^{10} + 54393900 p^{8} T^{12} - 2384 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 7634393359602348 p^{2} T^{9} + 72837035721440 p^{4} T^{10} - 615183153660 p^{6} T^{11} + 4512312618 p^{8} T^{12} - 27827520 p^{10} T^{13} + 137248 p^{12} T^{14} - 492 p^{14} T^{15} + p^{16} T^{16} \)
97 \( 1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} - 9811041724440 p^{4} T^{10} + 724155484 p^{8} T^{12} - 35880 p^{12} T^{14} + p^{16} T^{16} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.25259459094115082967270751852, −5.13543427818680149728389361528, −5.11578176935136806767891379815, −5.07912342495620898077409819724, −5.05206445565510467516714231971, −4.39680578299410982993962307567, −4.14854185842066092438285550327, −3.89935878444923605713000886653, −3.79882969927901047404357595070, −3.69053364202186744405003327083, −3.68229010343811104377055716010, −3.56764815892428971986028067582, −3.21869694681194812059058327139, −3.21461877769139074753383182254, −2.89801538916378466278269402778, −2.84968329835101944986895668365, −2.78725387599438033204172849071, −2.67381194644828292537135172196, −2.10498006632038874405151502940, −1.77320143551154581594278433126, −1.47281166997381092443151912662, −1.41282984720215686753738683983, −1.32873314219706409686147245452, −0.76394314042301869912600220836, −0.70644881144069948566373343580, 0.70644881144069948566373343580, 0.76394314042301869912600220836, 1.32873314219706409686147245452, 1.41282984720215686753738683983, 1.47281166997381092443151912662, 1.77320143551154581594278433126, 2.10498006632038874405151502940, 2.67381194644828292537135172196, 2.78725387599438033204172849071, 2.84968329835101944986895668365, 2.89801538916378466278269402778, 3.21461877769139074753383182254, 3.21869694681194812059058327139, 3.56764815892428971986028067582, 3.68229010343811104377055716010, 3.69053364202186744405003327083, 3.79882969927901047404357595070, 3.89935878444923605713000886653, 4.14854185842066092438285550327, 4.39680578299410982993962307567, 5.05206445565510467516714231971, 5.07912342495620898077409819724, 5.11578176935136806767891379815, 5.13543427818680149728389361528, 5.25259459094115082967270751852

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

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