Properties

Label 16-690e8-1.1-c1e8-0-0
Degree $16$
Conductor $5.138\times 10^{22}$
Sign $1$
Analytic cond. $849199.$
Root an. cond. $2.34727$
Motivic weight $1$
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  − 8·3-s + 32·9-s − 8·13-s − 2·16-s + 4·25-s − 88·27-s + 64·39-s − 16·43-s + 16·48-s − 32·61-s − 48·67-s + 24·73-s − 32·75-s + 206·81-s + 16·97-s − 48·103-s − 256·117-s − 40·121-s + 127-s + 128·129-s + 131-s + 137-s + 139-s − 64·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4.61·3-s + 32/3·9-s − 2.21·13-s − 1/2·16-s + 4/5·25-s − 16.9·27-s + 10.2·39-s − 2.43·43-s + 2.30·48-s − 4.09·61-s − 5.86·67-s + 2.80·73-s − 3.69·75-s + 22.8·81-s + 1.62·97-s − 4.72·103-s − 23.6·117-s − 3.63·121-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(849199.\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{690} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02017558202\)
\(L(\frac12)\) \(\approx\) \(0.02017558202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
5 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 4 T + 8 T^{2} + 12 T^{3} - 82 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 178 T^{4} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + 28 T^{2} + 342 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 8 T + 32 T^{2} + 24 T^{3} - 1582 T^{4} + 24 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 644 T^{4} + 3172230 T^{8} + 644 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 + 4828 T^{4} + 13645734 T^{8} + 4828 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 + 136 T^{2} + 11202 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 24 T + 288 T^{2} + 2184 T^{3} + 15986 T^{4} + 2184 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 32 T^{2} + 2562 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 12 T + 72 T^{2} - 516 T^{3} + 2798 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 296 T^{2} + 34290 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 12466 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 268 T^{2} + 32262 T^{4} + 268 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 8 T + 32 T^{2} - 744 T^{3} + 17282 T^{4} - 744 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.70065284424148711728586149022, −4.58654977757880897072969566590, −4.51108066876830105169121974798, −4.43861582749252643503474769174, −4.20859796228025181840564426864, −3.74620680755994955245182727496, −3.72331903374651836002115399120, −3.71465378532330843198774169791, −3.66168473135700658217784601471, −3.41805821829252120352112090655, −3.01827393744987360127559531679, −2.92749906540372938932034199923, −2.75950670067094563259586136731, −2.74252274766563989410714641979, −2.37235773137567227846136848616, −2.33255341142453757616483201392, −2.20774796845626966531739897226, −1.58628915072729756364566966460, −1.54938482530216525334286388301, −1.46267551943397379036372349319, −1.43396195901344587719508504781, −1.08388112414309498813505700366, −0.57248737610152807230572315066, −0.23432258351520421328157749539, −0.12463525596437182630034052012, 0.12463525596437182630034052012, 0.23432258351520421328157749539, 0.57248737610152807230572315066, 1.08388112414309498813505700366, 1.43396195901344587719508504781, 1.46267551943397379036372349319, 1.54938482530216525334286388301, 1.58628915072729756364566966460, 2.20774796845626966531739897226, 2.33255341142453757616483201392, 2.37235773137567227846136848616, 2.74252274766563989410714641979, 2.75950670067094563259586136731, 2.92749906540372938932034199923, 3.01827393744987360127559531679, 3.41805821829252120352112090655, 3.66168473135700658217784601471, 3.71465378532330843198774169791, 3.72331903374651836002115399120, 3.74620680755994955245182727496, 4.20859796228025181840564426864, 4.43861582749252643503474769174, 4.51108066876830105169121974798, 4.58654977757880897072969566590, 4.70065284424148711728586149022

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

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