Properties

Label 16-12e24-1.1-c2e8-0-1
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  − 48·13-s + 84·25-s + 160·37-s − 212·49-s + 32·61-s + 168·73-s + 216·97-s + 96·109-s + 68·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 808·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.69·13-s + 3.35·25-s + 4.32·37-s − 4.32·49-s + 0.524·61-s + 2.30·73-s + 2.22·97-s + 0.880·109-s + 0.561·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.78·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1291953631\)
\(L(\frac12)\) \(\approx\) \(0.1291953631\)
\(L(2)\) 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 \( ( 1 - 42 T^{2} + 59 p^{2} T^{4} - 42 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 + 106 T^{2} + 5667 T^{4} + 106 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 34 T^{2} + 21795 T^{4} - 34 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 12 T + 158 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
17 \( ( 1 - 876 T^{2} + 345062 T^{4} - 876 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 436 T^{2} + 183750 T^{4} + 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 1396 T^{2} + 922470 T^{4} - 1396 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 2892 T^{2} + 3450182 T^{4} - 2892 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 442 T^{2} + 478707 T^{4} + 442 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 40 T + 1194 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 2970 T^{2} + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 124 T^{2} + 5317350 T^{4} + 124 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 5308 T^{2} + 13692678 T^{4} - 5308 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 6858 T^{2} + 27521507 T^{4} - 6858 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 8668 T^{2} + 36921894 T^{4} - 8668 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 4 T + p^{2} T^{2} )^{8} \)
67 \( ( 1 + 5500 T^{2} + 9762342 T^{4} + 5500 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 16204 T^{2} + 114941670 T^{4} - 16204 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 42 T + 10883 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 + 2284 T^{2} - 11494938 T^{4} + 2284 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 11986 T^{2} + 124293075 T^{4} - 11986 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 15324 T^{2} + 123057542 T^{4} - 15324 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 27 T + p^{2} T^{2} )^{8} \)
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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

−3.71411684996642483289757203794, −3.48117878524557060024389584360, −3.39760704051869694628612351890, −3.37035187518789712285035059017, −3.20527292750213182473308051258, −3.05869068039332385126096807816, −3.00913305813259372728905213393, −2.69616659019086091228299412168, −2.59468564125321312912494968264, −2.52020482393453193336510974908, −2.44769000133787365613452589839, −2.39631378264428614320485093558, −2.21547281980588922015142180887, −2.20202348068984787004493138438, −1.84231609564529281348556881312, −1.73995600075720612564854466915, −1.53386171488895436323060172123, −1.20866394708854336019634373176, −1.19585967816687403301798918515, −1.16925449538886481294734328521, −0.809483728084877836965253625909, −0.60412197377942042526976511238, −0.57698040855371956292582442167, −0.31967650408057140921251550541, −0.02578625613428767159715936349, 0.02578625613428767159715936349, 0.31967650408057140921251550541, 0.57698040855371956292582442167, 0.60412197377942042526976511238, 0.809483728084877836965253625909, 1.16925449538886481294734328521, 1.19585967816687403301798918515, 1.20866394708854336019634373176, 1.53386171488895436323060172123, 1.73995600075720612564854466915, 1.84231609564529281348556881312, 2.20202348068984787004493138438, 2.21547281980588922015142180887, 2.39631378264428614320485093558, 2.44769000133787365613452589839, 2.52020482393453193336510974908, 2.59468564125321312912494968264, 2.69616659019086091228299412168, 3.00913305813259372728905213393, 3.05869068039332385126096807816, 3.20527292750213182473308051258, 3.37035187518789712285035059017, 3.39760704051869694628612351890, 3.48117878524557060024389584360, 3.71411684996642483289757203794

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

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