Properties

Degree 8
Conductor $ 2^{16} \cdot 5^{4} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.14e3·5-s + 4.51e4·9-s − 1.09e5·11-s + 6.36e5·19-s − 1.91e4·25-s − 3.53e6·29-s + 1.05e7·31-s − 1.67e7·41-s + 5.15e7·45-s + 5.72e7·49-s − 1.25e8·55-s + 4.60e8·59-s + 3.60e8·61-s + 4.76e7·71-s + 7.28e8·79-s + 9.91e8·81-s − 1.58e9·89-s + 7.26e8·95-s − 4.96e9·99-s + 2.27e9·101-s − 2.13e9·109-s − 1.29e9·121-s + 7.01e8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.815·5-s + 2.29·9-s − 2.26·11-s + 1.12·19-s − 0.00980·25-s − 0.927·29-s + 2.05·31-s − 0.927·41-s + 1.87·45-s + 1.41·49-s − 1.84·55-s + 4.95·59-s + 3.33·61-s + 0.222·71-s + 2.10·79-s + 2.56·81-s − 2.67·89-s + 0.914·95-s − 5.19·99-s + 2.17·101-s − 1.44·109-s − 0.547·121-s + 0.256·125-s − 0.756·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{80} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 40960000,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)$
$L(5)$  $\approx$  $7.55353$
$L(\frac12)$  $\approx$  $7.55353$
$L(\frac{11}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$D_{4}$ \( 1 - 228 p T + 422 p^{5} T^{2} - 228 p^{10} T^{3} + p^{18} T^{4} \)
good3$C_2^2 \wr C_2$ \( 1 - 5020 p^{2} T^{2} + 12953638 p^{4} T^{4} - 5020 p^{20} T^{6} + p^{36} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 166900 p^{3} T^{2} + 1461117678198 p^{4} T^{4} - 166900 p^{21} T^{6} + p^{36} T^{8} \)
11$D_{4}$ \( ( 1 + 54984 T + 5180465446 T^{2} + 54984 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - 35613791860 p^{18} T^{6} + p^{36} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - 285780369220 p^{18} T^{6} + p^{36} T^{8} \)
19$D_{4}$ \( ( 1 - 16760 p T + 505756418358 T^{2} - 16760 p^{10} T^{3} + p^{18} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 5779790962540 p^{18} T^{6} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 + 1765860 T + 26950935551038 T^{2} + 1765860 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 5293856 T + 59464921598526 T^{2} - 5293856 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
37$C_2^2 \wr C_2$ \( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - 231603274936660 p^{18} T^{6} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 8394276 T + 221313076168966 T^{2} + 8394276 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - 614109141147100 p^{18} T^{6} + p^{36} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - 1368976020813580 p^{18} T^{6} + p^{36} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - 7684297973864980 p^{18} T^{6} + p^{36} T^{8} \)
59$D_{4}$ \( ( 1 - 230414520 T + 28555631923987078 T^{2} - 230414520 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 180245284 T + 30154717014478446 T^{2} - 180245284 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + 41160407446058180 p^{18} T^{6} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 - 23805936 T + 85782754020107086 T^{2} - 23805936 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 - 229489314868712740 T^{2} + \)\(37\!\cdots\!22\)\( p^{2} T^{4} - 229489314868712740 p^{18} T^{6} + p^{36} T^{8} \)
79$D_{4}$ \( ( 1 - 364021760 T + 220545463862625438 T^{2} - 364021760 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 434569632367965820 p^{18} T^{6} + p^{36} T^{8} \)
89$D_{4}$ \( ( 1 + 791350380 T + 832192702699668118 T^{2} + 791350380 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
97$C_2^2 \wr C_2$ \( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - 2561123777205326980 p^{18} T^{6} + p^{36} T^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.462707599255700230685914409530, −8.442228961069140197960821438901, −8.232749292513288873417539740518, −7.63861570945178632431466401726, −7.50416167251085573976948489270, −7.33394130276484547415730213514, −6.68872787780404691940798313582, −6.68369588665202978028425921971, −6.51736075083567173522745182658, −5.61154466014508570380407418148, −5.33897631880823323120380488811, −5.28945617470984078496631646545, −5.24430808831981943073295864868, −4.38260877871394258121937534592, −4.28525216618922610783559680238, −3.84334344800049004983379725963, −3.50421662057204139631572919415, −2.98790541399688131350941171475, −2.34224561462314009350880672692, −2.28692208723966265863125018409, −2.15830639741145413924932622528, −1.29040815091873406371963258123, −1.12537785955381932095631722663, −0.74255264177482963577150369892, −0.35934690874260760425372986619, 0.35934690874260760425372986619, 0.74255264177482963577150369892, 1.12537785955381932095631722663, 1.29040815091873406371963258123, 2.15830639741145413924932622528, 2.28692208723966265863125018409, 2.34224561462314009350880672692, 2.98790541399688131350941171475, 3.50421662057204139631572919415, 3.84334344800049004983379725963, 4.28525216618922610783559680238, 4.38260877871394258121937534592, 5.24430808831981943073295864868, 5.28945617470984078496631646545, 5.33897631880823323120380488811, 5.61154466014508570380407418148, 6.51736075083567173522745182658, 6.68369588665202978028425921971, 6.68872787780404691940798313582, 7.33394130276484547415730213514, 7.50416167251085573976948489270, 7.63861570945178632431466401726, 8.232749292513288873417539740518, 8.442228961069140197960821438901, 8.462707599255700230685914409530

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