Properties

Degree 16
Conductor $ 2^{16} \cdot 3^{24} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 8·13-s + 16·16-s − 88·25-s − 56·37-s + 140·49-s + 16·52-s + 40·61-s − 88·64-s + 232·73-s − 248·97-s + 176·100-s − 560·109-s + 248·121-s + 127-s + 131-s + 137-s + 139-s + 112·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 596·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.615·13-s + 16-s − 3.51·25-s − 1.51·37-s + 20/7·49-s + 4/13·52-s + 0.655·61-s − 1.37·64-s + 3.17·73-s − 2.55·97-s + 1.75·100-s − 5.13·109-s + 2.04·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.756·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.52·169-s + 0.00578·173-s + 0.00558·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{16} \cdot 3^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{16} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.356776\)
\(L(\frac12)\)  \(\approx\)  \(0.356776\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 + p T^{2} - 3 p^{2} T^{4} + p^{5} T^{6} + p^{8} T^{8} \)
3 \( 1 \)
good5 \( ( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 - 10 p T^{2} + 5307 T^{4} - 10 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 124 T^{2} + 15126 T^{4} - 124 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 2 T + 159 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
17 \( ( 1 + 140 T^{2} + 136662 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 695 T^{2} + p^{4} T^{4} )^{4} \)
23 \( ( 1 - 604 T^{2} + 632886 T^{4} - 604 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 2468 T^{2} + 2752998 T^{4} + 2468 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 1540 T^{2} + 1702662 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 14 T + 2607 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 + 3140 T^{2} + 5167302 T^{4} + 3140 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 4516 T^{2} + 10784166 T^{4} - 4516 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 4444 T^{2} + 14436726 T^{4} - 4444 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 508 T^{2} + 14912358 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 6076 T^{2} + 23942166 T^{4} - 6076 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 10 T + 7287 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
67 \( ( 1 - 8110 T^{2} + 40110387 T^{4} - 8110 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 292 T^{2} + 42446598 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 58 T + 10779 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 934 T^{2} - 28603749 T^{4} - 934 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 26116 T^{2} + 265140006 T^{4} - 26116 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 + 30668 T^{2} + 360580758 T^{4} + 30668 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 62 T + 8259 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−6.02973085528930368936697458476, −5.77992919134299362254660276308, −5.75620782509359090412196574117, −5.71535192076209459497135949131, −5.57929281644472085540029494791, −5.37641742250104891724606986686, −4.95919035609663264046486485897, −4.89645823495769108944884924577, −4.78679579013376341227381725630, −4.73643627113021428583584722420, −4.12254795745738585339832000858, −3.99940642699208796017580529048, −3.96386930486340945850207757335, −3.80552935146742912679130076094, −3.65476343002419862204011902697, −3.44725267755043768281430803480, −3.10074580534916915835852164597, −2.57708406636371619572450183894, −2.56881973252412563258432380126, −2.47686259205557726263471131310, −1.85612662493467579453232115499, −1.84583082074051754470907594465, −1.32552692312873136290682505424, −0.906628958824251358634793561754, −0.15036262227245284629706915437, 0.15036262227245284629706915437, 0.906628958824251358634793561754, 1.32552692312873136290682505424, 1.84583082074051754470907594465, 1.85612662493467579453232115499, 2.47686259205557726263471131310, 2.56881973252412563258432380126, 2.57708406636371619572450183894, 3.10074580534916915835852164597, 3.44725267755043768281430803480, 3.65476343002419862204011902697, 3.80552935146742912679130076094, 3.96386930486340945850207757335, 3.99940642699208796017580529048, 4.12254795745738585339832000858, 4.73643627113021428583584722420, 4.78679579013376341227381725630, 4.89645823495769108944884924577, 4.95919035609663264046486485897, 5.37641742250104891724606986686, 5.57929281644472085540029494791, 5.71535192076209459497135949131, 5.75620782509359090412196574117, 5.77992919134299362254660276308, 6.02973085528930368936697458476

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

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