↑

Properties

Label 16-3060e8-1.1-c0e8-0-5
Degree $16$
Conductor $7.687\times 10^{27}$
Sign $1$
Analytic cond. $29.5820$
Root an. cond. $1.23577$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 16-s − 8·19-s − 4·29-s + 4·41-s − 4·59-s + 4·61-s − 8·71-s + 81-s − 8·89-s − 4·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 16-s − 8·19-s − 4·29-s + 4·41-s − 4·59-s + 4·61-s − 8·71-s + 81-s − 8·89-s − 4·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(29.5820\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2644485441\)
\(L(\frac12)\) \(\approx\) \(0.2644485441\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 - T^{4} + T^{8} \)
5 \( 1 - T^{4} + T^{8} \)
17 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T^{4} + T^{8} )^{2} \)
13 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( ( 1 + T + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
31 \( ( 1 - T^{4} + T^{8} )^{2} \)
37 \( ( 1 - T^{4} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} )^{4} \)
53 \( ( 1 + T^{2} )^{8} \)
59 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} )^{4} \)
71 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 - T^{4} + T^{8} )^{2} \)
83 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
89 \( ( 1 + T + T^{2} )^{8} \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.02287619792768723994201795018, −3.92519966544518722581278113995, −3.83314985445791480478721445586, −3.42488852535591619105546401148, −3.37974268196763247247202877455, −3.15692958819277526165467486360, −2.98908690665198716527064080780, −2.91917291962515390800298208730, −2.78638953983634320453937175440, −2.76993428378933544429053190868, −2.75207187344942004821548143358, −2.64886931949734246548117315045, −2.36692942837220671111382962500, −2.13572873324579691924442833067, −2.04482797511391865026973160758, −1.88228741637393717897901458578, −1.85144890231971103490841204050, −1.78776664950927118286599439197, −1.66291503669152745347669707616, −1.54114097863756942579956503442, −1.37952753666690062054914334015, −1.16117107179965856398044851143, −0.75274853330840981537794667153, −0.43672668664789139288285486945, −0.21545597317553655936729959504, 0.21545597317553655936729959504, 0.43672668664789139288285486945, 0.75274853330840981537794667153, 1.16117107179965856398044851143, 1.37952753666690062054914334015, 1.54114097863756942579956503442, 1.66291503669152745347669707616, 1.78776664950927118286599439197, 1.85144890231971103490841204050, 1.88228741637393717897901458578, 2.04482797511391865026973160758, 2.13572873324579691924442833067, 2.36692942837220671111382962500, 2.64886931949734246548117315045, 2.75207187344942004821548143358, 2.76993428378933544429053190868, 2.78638953983634320453937175440, 2.91917291962515390800298208730, 2.98908690665198716527064080780, 3.15692958819277526165467486360, 3.37974268196763247247202877455, 3.42488852535591619105546401148, 3.83314985445791480478721445586, 3.92519966544518722581278113995, 4.02287619792768723994201795018

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

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