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  − 4·4-s + 4·7-s + 10·16-s − 16·28-s + 8·37-s + 8·41-s + 16·43-s − 40·47-s + 10·49-s − 20·64-s − 32·67-s + 8·79-s + 16·83-s + 8·89-s − 8·101-s − 32·109-s + 40·112-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s + 1.51·7-s + 5/2·16-s − 3.02·28-s + 1.31·37-s + 1.24·41-s + 2.43·43-s − 5.83·47-s + 10/7·49-s − 5/2·64-s − 3.90·67-s + 0.900·79-s + 1.75·83-s + 0.847·89-s − 0.796·101-s − 3.06·109-s + 3.77·112-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.5411269944$
$L(\frac12)$  $\approx$  $0.5411269944$
$L(\frac{3}{2})$   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 \( ( 1 + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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 \( 1 - 44 T^{2} + 872 T^{4} - 756 p T^{6} + 102190 T^{8} - 756 p^{3} T^{10} + 872 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 18 T^{2} + 120 T^{3} - 38 T^{4} + 120 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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 \( 1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 123312 p^{2} T^{10} + 3740 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 - 4 T + 84 T^{2} - 532 T^{3} + 3602 T^{4} - 532 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( ( 1 - 8 T + 146 T^{2} - 984 T^{3} + 8842 T^{4} - 984 p T^{5} + 146 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 20 T + 166 T^{2} + 396 T^{3} - 838 T^{4} + 396 p T^{5} + 166 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 3946320 p^{2} T^{10} + 43772 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 + 16 T + 218 T^{2} + 1728 T^{3} + 15338 T^{4} + 1728 p T^{5} + 218 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 3074136 p^{2} T^{10} + 30332 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 - 8 T + 244 T^{2} - 1800 T^{3} + 27878 T^{4} - 1800 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 4059696 p^{2} T^{10} + 38300 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.57723800880555471974376389807, −3.38254574773465320517974822313, −3.37121842645654013208745031075, −3.32997362348800033108465860902, −3.19193278162007376066262115585, −3.09557148168930825679279819238, −2.84676147510146393778375151789, −2.82855314870373420254563884319, −2.61474431480736526185906587950, −2.51523141336406952461333943888, −2.47447128797894892784832642815, −2.27895281693706704403938940712, −2.06238771837244040127523595561, −1.91373628530932425692672009455, −1.80928622493081156714518889660, −1.60369212421406493496746736545, −1.59462041276080583533582226366, −1.48159398043644526969639088222, −1.22459939472148983396711582909, −1.08898612107020500047483447233, −0.792558116109955453680233835246, −0.77581256488359184514870544512, −0.76447652619687584182515653060, −0.17738316155944236902699729960, −0.12268417269414241463177502929, 0.12268417269414241463177502929, 0.17738316155944236902699729960, 0.76447652619687584182515653060, 0.77581256488359184514870544512, 0.792558116109955453680233835246, 1.08898612107020500047483447233, 1.22459939472148983396711582909, 1.48159398043644526969639088222, 1.59462041276080583533582226366, 1.60369212421406493496746736545, 1.80928622493081156714518889660, 1.91373628530932425692672009455, 2.06238771837244040127523595561, 2.27895281693706704403938940712, 2.47447128797894892784832642815, 2.51523141336406952461333943888, 2.61474431480736526185906587950, 2.82855314870373420254563884319, 2.84676147510146393778375151789, 3.09557148168930825679279819238, 3.19193278162007376066262115585, 3.32997362348800033108465860902, 3.37121842645654013208745031075, 3.38254574773465320517974822313, 3.57723800880555471974376389807

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

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