Properties

Label 16-12e24-1.1-c3e8-0-4
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.16755\times 10^{16}$
Root an. cond. $10.0972$
Motivic weight $3$
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  + 284·25-s + 940·49-s − 1.28e3·73-s + 2.29e3·97-s + 5.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.75e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 2.27·25-s + 2.74·49-s − 2.06·73-s + 2.40·97-s + 3.95·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.16755\times 10^{16}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(14.02114387\)
\(L(\frac12)\) \(\approx\) \(14.02114387\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 142 T^{2} + 4539 T^{4} - 142 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
7 \( ( 1 - 470 T^{2} + 103251 T^{4} - 470 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 - 126 T + 6623 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} )^{2}( 1 + 126 T + 6623 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13 \( ( 1 - p^{3} T^{2} )^{8} \)
17 \( ( 1 - p^{3} T^{2} )^{8} \)
19 \( ( 1 + p^{3} T^{2} )^{8} \)
23 \( ( 1 + p^{3} T^{2} )^{8} \)
29 \( ( 1 + 1150 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 + 54682 T^{2} + 2102617443 T^{4} + 54682 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - p^{3} T^{2} )^{8} \)
41 \( ( 1 - p^{3} T^{2} )^{8} \)
43 \( ( 1 + p^{3} T^{2} )^{8} \)
47 \( ( 1 + p^{3} T^{2} )^{8} \)
53 \( ( 1 - 38446 T^{2} - 20686266213 T^{4} - 38446 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 + 103430 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - p^{3} T^{2} )^{8} \)
67 \( ( 1 + p^{3} T^{2} )^{8} \)
71 \( ( 1 + p^{3} T^{2} )^{8} \)
73 \( ( 1 + 322 T - 285333 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
79 \( ( 1 + 890822 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 1530 T + 1352087 T^{2} - 1530 p^{3} T^{3} + p^{6} T^{4} )^{2}( 1 + 1530 T + 1352087 T^{2} + 1530 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89 \( ( 1 - p^{3} T^{2} )^{8} \)
97 \( ( 1 - 574 T - 583197 T^{2} - 574 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
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

−3.53771023620112635753012448274, −3.47191074380228255829273700163, −3.36731455993187816006000070711, −3.00608745918441258100371262923, −3.00537923533238538888345473837, −2.96302846257440436426636722083, −2.80710971356938535959448214209, −2.55518823318183839013848617267, −2.54646805351729937467679154065, −2.45621567536351298569750852270, −2.26795281373470640365500841564, −1.97983552459157085337686332843, −1.91250487331663417477359318392, −1.91124110912727304608270272615, −1.80250136947160897355927547856, −1.47240105764935653483230976916, −1.30513344451521994205020323121, −1.16715199747438864154852866651, −1.10662528143205462474768873094, −0.882308007455229233761018759475, −0.71050355990875859017752679713, −0.67501111275389604850478913148, −0.46315754505893177143653726586, −0.28145028907757604058026380581, −0.18511596569642236505261632756, 0.18511596569642236505261632756, 0.28145028907757604058026380581, 0.46315754505893177143653726586, 0.67501111275389604850478913148, 0.71050355990875859017752679713, 0.882308007455229233761018759475, 1.10662528143205462474768873094, 1.16715199747438864154852866651, 1.30513344451521994205020323121, 1.47240105764935653483230976916, 1.80250136947160897355927547856, 1.91124110912727304608270272615, 1.91250487331663417477359318392, 1.97983552459157085337686332843, 2.26795281373470640365500841564, 2.45621567536351298569750852270, 2.54646805351729937467679154065, 2.55518823318183839013848617267, 2.80710971356938535959448214209, 2.96302846257440436426636722083, 3.00537923533238538888345473837, 3.00608745918441258100371262923, 3.36731455993187816006000070711, 3.47191074380228255829273700163, 3.53771023620112635753012448274

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

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