↑

Properties

Label 8-12e4-1.1-c16e4-0-0
Degree $8$
Conductor $20736$
Sign $1$
Analytic cond. $143966.$
Root an. cond. $4.41349$
Motivic weight $16$
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  − 1.24e4·3-s − 1.05e6·7-s + 7.20e7·9-s − 2.74e8·13-s − 4.13e10·19-s + 1.30e10·21-s + 4.23e10·25-s − 4.08e11·27-s − 1.71e12·31-s + 5.25e12·37-s + 3.41e12·39-s + 1.89e13·43-s − 9.14e13·49-s + 5.13e14·57-s − 9.84e14·61-s − 7.56e13·63-s − 7.98e14·67-s − 2.12e15·73-s − 5.26e14·75-s − 1.94e15·79-s + 3.94e15·81-s + 2.88e14·91-s + 2.12e16·93-s − 3.18e16·97-s + 3.52e16·103-s + 3.43e16·109-s − 6.53e16·111-s + ⋯
L(s)  = 1  − 1.89·3-s − 0.182·7-s + 1.67·9-s − 0.336·13-s − 2.43·19-s + 0.344·21-s + 0.277·25-s − 1.44·27-s − 2.00·31-s + 1.49·37-s + 0.637·39-s + 1.62·43-s − 2.75·49-s + 4.60·57-s − 5.13·61-s − 0.305·63-s − 1.96·67-s − 2.63·73-s − 0.525·75-s − 1.28·79-s + 2.12·81-s + 0.0613·91-s + 3.80·93-s − 4.06·97-s + 2.78·103-s + 1.72·109-s − 2.83·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(143966.\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 20736,\ (\ :8, 8, 8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.3308975903\)
\(L(\frac12)\) \(\approx\) \(0.3308975903\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$D_{4}$ \( 1 + 460 p^{3} T + 112742 p^{6} T^{2} + 460 p^{19} T^{3} + p^{32} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 8471280596 p T^{2} - 8952186757218521874 p^{5} T^{4} - 8471280596 p^{33} T^{6} + p^{64} T^{8} \)
7$D_{4}$ \( ( 1 + 75020 p T + 134516880282 p^{3} T^{2} + 75020 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 12354555851874284 p T^{2} + \)\(69\!\cdots\!66\)\( p^{2} T^{4} - 12354555851874284 p^{33} T^{6} + p^{64} T^{8} \)
13$D_{4}$ \( ( 1 + 10571660 p T + 5498519503620054 p^{2} T^{2} + 10571660 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 66674679099167946244 T^{2} + \)\(56\!\cdots\!06\)\( T^{4} - 66674679099167946244 p^{32} T^{6} + p^{64} T^{8} \)
19$D_{4}$ \( ( 1 + 20674369892 T + \)\(68\!\cdots\!78\)\( T^{2} + 20674369892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + \)\(68\!\cdots\!76\)\( T^{2} + \)\(85\!\cdots\!86\)\( T^{4} + \)\(68\!\cdots\!76\)\( p^{32} T^{6} + p^{64} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(43\!\cdots\!04\)\( T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(43\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
31$D_{4}$ \( ( 1 + 856168243892 T + \)\(14\!\cdots\!78\)\( T^{2} + 856168243892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 2629335341860 T + \)\(23\!\cdots\!38\)\( T^{2} - 2629335341860 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(83\!\cdots\!84\)\( T^{2} + \)\(65\!\cdots\!26\)\( T^{4} - \)\(83\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \)
43$D_{4}$ \( ( 1 - 9485084864860 T + \)\(84\!\cdots\!86\)\( T^{2} - 9485084864860 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(20\!\cdots\!64\)\( T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(11\!\cdots\!64\)\( T^{2} + \)\(57\!\cdots\!06\)\( T^{4} - \)\(11\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(32\!\cdots\!44\)\( T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
61$D_{4}$ \( ( 1 + 492098526528092 T + \)\(13\!\cdots\!38\)\( T^{2} + 492098526528092 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 399005728500740 T + \)\(36\!\cdots\!86\)\( T^{2} + 399005728500740 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(70\!\cdots\!64\)\( T^{2} + \)\(38\!\cdots\!86\)\( p T^{4} - \)\(70\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
73$D_{4}$ \( ( 1 + 1064074159757180 T + \)\(15\!\cdots\!18\)\( T^{2} + 1064074159757180 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 974906968346036 T + \)\(21\!\cdots\!66\)\( T^{2} + 974906968346036 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(14\!\cdots\!44\)\( T^{2} + \)\(99\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + \)\(11\!\cdots\!36\)\( T^{2} + \)\(46\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!36\)\( p^{32} T^{6} + p^{64} T^{8} \)
97$D_{4}$ \( ( 1 + 15925984078821020 T + \)\(18\!\cdots\!06\)\( T^{2} + 15925984078821020 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.23525670741630531272604113547, −10.91033821211018912871389208438, −10.80928379262354413327803303357, −10.16893159440793796773345629121, −9.841385404372896903965478109580, −9.319402808242157274298555545634, −8.868249641037154626405428432784, −8.582082929427103613420358649584, −7.77969660179754618359833400286, −7.40350867043986406072618456250, −7.19836608238472308920156592652, −6.24285613932394715867556731260, −6.22964212069033499464745795887, −5.94172978267780186534261806589, −5.52602467919457870438717818383, −4.65810639867388815092374710489, −4.50741851147657480384502849419, −4.29780165744016960879897886365, −3.34132122187593453164746344300, −2.94228721146662208229852130339, −2.20415032095252508291977939000, −1.56576346920621534391179759837, −1.43479365768162528180342393087, −0.32081524903498784983758548370, −0.27783184393114284804012095406, 0.27783184393114284804012095406, 0.32081524903498784983758548370, 1.43479365768162528180342393087, 1.56576346920621534391179759837, 2.20415032095252508291977939000, 2.94228721146662208229852130339, 3.34132122187593453164746344300, 4.29780165744016960879897886365, 4.50741851147657480384502849419, 4.65810639867388815092374710489, 5.52602467919457870438717818383, 5.94172978267780186534261806589, 6.22964212069033499464745795887, 6.24285613932394715867556731260, 7.19836608238472308920156592652, 7.40350867043986406072618456250, 7.77969660179754618359833400286, 8.582082929427103613420358649584, 8.868249641037154626405428432784, 9.319402808242157274298555545634, 9.841385404372896903965478109580, 10.16893159440793796773345629121, 10.80928379262354413327803303357, 10.91033821211018912871389208438, 11.23525670741630531272604113547

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