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Properties

Label 16-799e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.661\times 10^{23}$
Sign $1$
Analytic cond. $0.000639191$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·4-s + 2·7-s + 2·9-s − 4·12-s + 16-s − 2·17-s − 4·21-s − 2·27-s + 4·28-s + 4·36-s + 2·37-s − 8·47-s − 2·48-s + 2·49-s + 4·51-s − 2·61-s + 4·63-s − 4·68-s − 2·71-s − 2·79-s + 81-s − 8·84-s − 4·89-s + 2·97-s − 4·108-s − 4·111-s + ⋯
L(s)  = 1  − 2·3-s + 2·4-s + 2·7-s + 2·9-s − 4·12-s + 16-s − 2·17-s − 4·21-s − 2·27-s + 4·28-s + 4·36-s + 2·37-s − 8·47-s − 2·48-s + 2·49-s + 4·51-s − 2·61-s + 4·63-s − 4·68-s − 2·71-s − 2·79-s + 81-s − 8·84-s − 4·89-s + 2·97-s − 4·108-s − 4·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(17^{8} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(0.000639191\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 17^{8} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4241431091\)
\(L(\frac12)\) \(\approx\) \(0.4241431091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47 \( ( 1 + T )^{8} \)
good2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
5 \( ( 1 + T^{4} )^{4} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
11 \( ( 1 + T^{4} )^{4} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 + T^{4} )^{4} \)
31 \( ( 1 + T^{4} )^{4} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−4.56047503026960774596711923914, −4.46247090255594849826074991755, −4.44045143367586847739925641252, −4.39969481491432142859437293153, −4.31041590630269292513213728810, −4.27425796892834831771647267274, −4.00011311388167499768079791635, −3.93264964012453810592051906430, −3.56346144148815042117010944740, −3.45622213338103050020346655998, −3.11060227590902797202070711377, −3.00094901969608208388685868842, −2.97073178173103081062108242035, −2.95224292272376695219224146868, −2.81346149624886806405363955457, −2.58582575141961639504201372891, −2.33726233986095704338060907860, −1.83704402982401397509200382824, −1.81603116458268595366090047535, −1.81084080736197655401449178659, −1.76000343888954358364343851002, −1.74748537697137233320452185668, −1.50021595385170185476727381398, −1.06632027335765596283969352211, −0.60800463655263510644104495712, 0.60800463655263510644104495712, 1.06632027335765596283969352211, 1.50021595385170185476727381398, 1.74748537697137233320452185668, 1.76000343888954358364343851002, 1.81084080736197655401449178659, 1.81603116458268595366090047535, 1.83704402982401397509200382824, 2.33726233986095704338060907860, 2.58582575141961639504201372891, 2.81346149624886806405363955457, 2.95224292272376695219224146868, 2.97073178173103081062108242035, 3.00094901969608208388685868842, 3.11060227590902797202070711377, 3.45622213338103050020346655998, 3.56346144148815042117010944740, 3.93264964012453810592051906430, 4.00011311388167499768079791635, 4.27425796892834831771647267274, 4.31041590630269292513213728810, 4.39969481491432142859437293153, 4.44045143367586847739925641252, 4.46247090255594849826074991755, 4.56047503026960774596711923914

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

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