Properties

Label 16-570e8-1.1-c1e8-0-2
Degree $16$
Conductor $1.114\times 10^{22}$
Sign $1$
Analytic cond. $184168.$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·4-s − 2·9-s + 16·12-s + 10·16-s − 8·19-s − 4·25-s + 40·27-s + 8·36-s − 16·37-s − 40·48-s − 12·49-s + 32·57-s + 8·61-s − 20·64-s − 56·67-s + 16·75-s + 32·76-s − 55·81-s − 80·97-s + 16·100-s − 8·103-s − 160·108-s + 64·111-s + 40·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2.30·3-s − 2·4-s − 2/3·9-s + 4.61·12-s + 5/2·16-s − 1.83·19-s − 4/5·25-s + 7.69·27-s + 4/3·36-s − 2.63·37-s − 5.77·48-s − 1.71·49-s + 4.23·57-s + 1.02·61-s − 5/2·64-s − 6.84·67-s + 1.84·75-s + 3.67·76-s − 6.11·81-s − 8.12·97-s + 8/5·100-s − 0.788·103-s − 15.3·108-s + 6.07·111-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(184168.\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.002871459802\)
\(L(\frac12)\) \(\approx\) \(0.002871459802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( ( 1 + T + p T^{2} )^{4} \)
5 \( 1 + 4 T^{2} + 2 p T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
19 \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
good7 \( ( 1 + 6 T^{2} + 9 p T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 20 T^{2} + 298 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 2 p T^{2} + 823 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 6 T^{2} + 1023 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 46 T^{2} + 1111 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 44 T^{2} + 1306 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T + p T^{2} )^{8} \)
41 \( ( 1 + 20 T^{2} - 102 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 156 T^{2} + 9738 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 84 T^{2} + 5478 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 105 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 106 T^{2} + 8671 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 2 T + 112 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 7 T + p T^{2} )^{8} \)
71 \( ( 1 + 60 T^{2} + 10938 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 166 T^{2} + 13983 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 60 T^{2} + 2118 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 196 T^{2} + 22678 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 276 T^{2} + 33786 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 20 T + 250 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{4} \)
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

−4.73958553200189097475567714687, −4.68844190599688906920407234780, −4.55964636885800844474610316316, −4.38481917683447837996262080359, −4.27984778381846457213778936028, −4.08237092302616647193570666039, −3.89975370412951660500079647512, −3.68887578599870942221811668942, −3.57579336663835477239546387646, −3.54703822026208896948739037395, −3.23542272071813065947121546218, −3.10238431273673592532334373646, −3.04632443277903002603449380817, −2.65190830678613713680380445554, −2.64768766831177788697972177238, −2.52622881324610911821651116256, −2.49410375228111921878876643898, −1.83438040352828041271059572408, −1.69075745186330752205149918914, −1.60666777510074368458970204794, −1.43489400938269139052240448467, −1.03292326427899549499258678015, −0.67689418862392362457379480049, −0.16860702723137635626030529504, −0.06491582616942036811841281575, 0.06491582616942036811841281575, 0.16860702723137635626030529504, 0.67689418862392362457379480049, 1.03292326427899549499258678015, 1.43489400938269139052240448467, 1.60666777510074368458970204794, 1.69075745186330752205149918914, 1.83438040352828041271059572408, 2.49410375228111921878876643898, 2.52622881324610911821651116256, 2.64768766831177788697972177238, 2.65190830678613713680380445554, 3.04632443277903002603449380817, 3.10238431273673592532334373646, 3.23542272071813065947121546218, 3.54703822026208896948739037395, 3.57579336663835477239546387646, 3.68887578599870942221811668942, 3.89975370412951660500079647512, 4.08237092302616647193570666039, 4.27984778381846457213778936028, 4.38481917683447837996262080359, 4.55964636885800844474610316316, 4.68844190599688906920407234780, 4.73958553200189097475567714687

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

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