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Properties

Label 16-3332e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $58.4651$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s − 4·73-s − 4·80-s + 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s − 4·73-s − 4·80-s + 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(58.4651\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4451083032\)
\(L(\frac12)\) \(\approx\) \(0.4451083032\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
7 \( 1 \)
17 \( 1 - T^{4} + T^{8} \)
good3 \( 1 - T^{8} + T^{16} \)
5 \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
11 \( 1 - T^{8} + T^{16} \)
13 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( 1 - T^{8} + T^{16} \)
29 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
31 \( 1 - T^{8} + T^{16} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} )^{4} \)
53 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
59 \( ( 1 - T^{4} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
79 \( 1 - T^{8} + T^{16} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.70711713474621701676871875243, −3.63811592093273672741225142519, −3.60273123175848643784808390330, −3.54622319367956861352127656597, −3.49869172565451016974559020648, −3.46595150318003168243900735760, −3.27012524113220729940719904255, −2.86033546862184859570663045655, −2.81929553358394672626968431942, −2.68506171603975328910482062844, −2.64217750023780438875714466586, −2.46942171609113272626444025911, −2.44780871946282110045119261734, −2.33937338753010755157741205633, −2.33148123389208182931169255910, −1.97444567019157911734429425026, −1.79088260996609759252790880776, −1.51601818799201995453862550902, −1.46801307288539836521436001588, −1.24184580967821374918365488716, −1.09655978153054501608470272356, −1.05036628784670315363344964580, −0.789264978740373632525103578207, −0.52099088883558125701617172638, −0.30312208883794207120868664705, 0.30312208883794207120868664705, 0.52099088883558125701617172638, 0.789264978740373632525103578207, 1.05036628784670315363344964580, 1.09655978153054501608470272356, 1.24184580967821374918365488716, 1.46801307288539836521436001588, 1.51601818799201995453862550902, 1.79088260996609759252790880776, 1.97444567019157911734429425026, 2.33148123389208182931169255910, 2.33937338753010755157741205633, 2.44780871946282110045119261734, 2.46942171609113272626444025911, 2.64217750023780438875714466586, 2.68506171603975328910482062844, 2.81929553358394672626968431942, 2.86033546862184859570663045655, 3.27012524113220729940719904255, 3.46595150318003168243900735760, 3.49869172565451016974559020648, 3.54622319367956861352127656597, 3.60273123175848643784808390330, 3.63811592093273672741225142519, 3.70711713474621701676871875243

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

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