Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \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  + 8·2-s + 36·4-s + 120·8-s − 8·13-s + 330·16-s − 8·23-s − 64·26-s + 792·32-s − 8·41-s − 64·46-s − 2·49-s − 288·52-s − 8·53-s + 1.71e3·64-s + 48·73-s − 8·79-s − 64·82-s + 8·89-s − 288·92-s − 24·97-s − 16·98-s + 8·101-s − 56·103-s − 960·104-s − 64·106-s + 40·107-s + 32·109-s + ⋯
L(s)  = 1  + 5.65·2-s + 18·4-s + 42.4·8-s − 2.21·13-s + 82.5·16-s − 1.66·23-s − 12.5·26-s + 140.·32-s − 1.24·41-s − 9.43·46-s − 2/7·49-s − 39.9·52-s − 1.09·53-s + 214.5·64-s + 5.61·73-s − 0.900·79-s − 7.06·82-s + 0.847·89-s − 30.0·92-s − 2.43·97-s − 1.61·98-s + 0.796·101-s − 5.51·103-s − 94.1·104-s − 6.21·106-s + 3.86·107-s + 3.06·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $362.2929778$
$L(\frac12)$  $\approx$  $362.2929778$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) 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 \( ( 1 - T )^{8} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \)
good11 \( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 11916 p^{2} T^{10} + 776 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 4 T + 30 T^{2} + 132 T^{3} + 514 T^{4} + 132 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 5364 p^{2} T^{10} + 248 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 31140 p^{2} T^{10} + 1256 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 4 T + 48 T^{2} + 180 T^{3} + 1438 T^{4} + 180 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 399744 p^{2} T^{10} + 8732 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21984 p^{2} T^{10} + 572 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 394632 p^{2} T^{10} + 10004 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 468 p T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 1476524 p^{2} T^{10} + 23256 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 68 T^{2} + 10040 T^{4} + 9492 p T^{6} + 34404910 T^{8} + 9492 p^{3} T^{10} + 10040 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 4 T + 160 T^{2} + 12 p T^{3} + 11630 T^{4} + 12 p^{2} T^{5} + 160 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 312 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 2804772 p^{2} T^{10} + 31208 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 1953756 p^{2} T^{10} + 22904 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 12828132 p^{2} T^{10} + 98600 p^{4} T^{12} - 460 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 24 T + 404 T^{2} - 4680 T^{3} + 45878 T^{4} - 4680 p T^{5} + 404 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 532 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 11335896 p^{2} T^{10} + 86492 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 1308 p T^{5} + 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 12 T + 176 T^{2} + 2148 T^{3} + 29438 T^{4} + 2148 p T^{5} + 176 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.71192994878061393164178472531, −3.44741772341814301699968409238, −3.41567140822493520066403196785, −3.23896856077855770824196658478, −3.20966700973152877416588310420, −3.20929636462895420238120967839, −3.10166076847864589433293991528, −2.83566115920167901717179092366, −2.73353709572955696369996338504, −2.60353331875382842459920240141, −2.48258886580716257080643522384, −2.35718366561643472085788156160, −2.22750443093247662853771886187, −2.15828539523710888270885430788, −2.10092221341380082174925806623, −1.92877478671784693149104763058, −1.84498342170850924875159663402, −1.79604950851459884046234479822, −1.40105246598310387289546053170, −1.33592873431003705149913965815, −1.17364360427669570259429392107, −0.997346602136186534893804152744, −0.71474648593895066474248615188, −0.38656616488429029136967595805, −0.24417949452123263842519559640, 0.24417949452123263842519559640, 0.38656616488429029136967595805, 0.71474648593895066474248615188, 0.997346602136186534893804152744, 1.17364360427669570259429392107, 1.33592873431003705149913965815, 1.40105246598310387289546053170, 1.79604950851459884046234479822, 1.84498342170850924875159663402, 1.92877478671784693149104763058, 2.10092221341380082174925806623, 2.15828539523710888270885430788, 2.22750443093247662853771886187, 2.35718366561643472085788156160, 2.48258886580716257080643522384, 2.60353331875382842459920240141, 2.73353709572955696369996338504, 2.83566115920167901717179092366, 3.10166076847864589433293991528, 3.20929636462895420238120967839, 3.20966700973152877416588310420, 3.23896856077855770824196658478, 3.41567140822493520066403196785, 3.44741772341814301699968409238, 3.71192994878061393164178472531

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

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