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

Downloads

Learn more about

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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.008455109288$
$L(\frac12)$  $\approx$  $0.008455109288$
$L(\frac{3}{2})$   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 \( ( 1 - 12 T + 176 T^{2} - 2148 T^{3} + 29438 T^{4} - 2148 p T^{5} + 176 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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

−3.57491806773274905914523954273, −3.18201809179626461189918601469, −3.12560491307407065571191377562, −3.09079749823996720169277730001, −3.08271248630094405356581953395, −2.92090148488734252992407009456, −2.72368915643235189471755829396, −2.70130752775983548464646715240, −2.58582757916499255223281476691, −2.33781288695137898802605932851, −2.27448640888268070174712476174, −1.97833928939664303962700457567, −1.90439262616827055312852490495, −1.76655945762699630905003130484, −1.73712502031506145192815842108, −1.68798501433387629510545251105, −1.50932956047106054712998559153, −1.33062058394652764360774399003, −1.00960643775017485623493210670, −0.924805462773165325283736635366, −0.890999467844483435646953202729, −0.841240448747767722625190576284, −0.74364155357605945851176613420, −0.094276112353557039839528094392, −0.079188706485313008974897166107, 0.079188706485313008974897166107, 0.094276112353557039839528094392, 0.74364155357605945851176613420, 0.841240448747767722625190576284, 0.890999467844483435646953202729, 0.924805462773165325283736635366, 1.00960643775017485623493210670, 1.33062058394652764360774399003, 1.50932956047106054712998559153, 1.68798501433387629510545251105, 1.73712502031506145192815842108, 1.76655945762699630905003130484, 1.90439262616827055312852490495, 1.97833928939664303962700457567, 2.27448640888268070174712476174, 2.33781288695137898802605932851, 2.58582757916499255223281476691, 2.70130752775983548464646715240, 2.72368915643235189471755829396, 2.92090148488734252992407009456, 3.08271248630094405356581953395, 3.09079749823996720169277730001, 3.12560491307407065571191377562, 3.18201809179626461189918601469, 3.57491806773274905914523954273

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

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