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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.8265720314$
$L(\frac12)$  $\approx$  $0.8265720314$
$L(\frac{3}{2})$   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

−3.73642836469040030198739099510, −3.39315723838559373707505193266, −3.36873456936668321110709345846, −3.29866918555952496522794985029, −3.21153580476244868162254556021, −3.01653851854856646531370819900, −2.87680318727254087429251046962, −2.74184197014620646164422497747, −2.61777538240723846530650680493, −2.47786810874438573541069128311, −2.33860276417307073545031763832, −2.33137771344974794232911307186, −2.20612840731816362074763447110, −2.10566838868654507123939159297, −1.73607955160185689410687352916, −1.59180717863551160522424372429, −1.58940311492578859552958713557, −1.37632763665292059651761729974, −1.12061843740778628208891698941, −1.10403384854109550611903610918, −0.872262347607680052691396809200, −0.804549297199860772132410840058, −0.78102243317445115726269937994, −0.19873983003084499707183127570, −0.12878817138318499981805586129, 0.12878817138318499981805586129, 0.19873983003084499707183127570, 0.78102243317445115726269937994, 0.804549297199860772132410840058, 0.872262347607680052691396809200, 1.10403384854109550611903610918, 1.12061843740778628208891698941, 1.37632763665292059651761729974, 1.58940311492578859552958713557, 1.59180717863551160522424372429, 1.73607955160185689410687352916, 2.10566838868654507123939159297, 2.20612840731816362074763447110, 2.33137771344974794232911307186, 2.33860276417307073545031763832, 2.47786810874438573541069128311, 2.61777538240723846530650680493, 2.74184197014620646164422497747, 2.87680318727254087429251046962, 3.01653851854856646531370819900, 3.21153580476244868162254556021, 3.29866918555952496522794985029, 3.36873456936668321110709345846, 3.39315723838559373707505193266, 3.73642836469040030198739099510

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

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