Properties

Degree 8
Conductor $ 2^{16} \cdot 5^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 16·19-s + 8·25-s − 8·29-s − 16·31-s − 24·41-s − 2·49-s − 16·59-s + 24·61-s + 24·71-s + 8·79-s − 18·81-s + 40·89-s + 64·95-s + 24·101-s − 8·109-s + 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s − 64·155-s + 157-s + ⋯
L(s)  = 1  + 1.78·5-s + 3.67·19-s + 8/5·25-s − 1.48·29-s − 2.87·31-s − 3.74·41-s − 2/7·49-s − 2.08·59-s + 3.07·61-s + 2.84·71-s + 0.900·79-s − 2·81-s + 4.23·89-s + 6.56·95-s + 2.38·101-s − 0.766·109-s + 4/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{560} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{16} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $3.85155$
$L(\frac12)$  $\approx$  $3.85155$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$D_{4}$ \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.73995901752512594892029298067, −7.68414523634273407074751041424, −7.19258741600170091374277019536, −6.99415327141256090395497325092, −6.90815536370214047351725288857, −6.57452826734288802278571971606, −6.34011920926801764563699016675, −6.02786747564235330744470976082, −5.61574650205583044526235799684, −5.44633409098404133479948268008, −5.40937285065601616465906456078, −5.25152056206810604680905359809, −4.99537457163729420102410260742, −4.65498515269625819021756396253, −4.29419513522135571983399784796, −3.56234497325956004639105698721, −3.48738172840794368963940001970, −3.40736505686345329035316448916, −3.30188167355794241111823405207, −2.62398767237310129532409966925, −2.12770916693379347348022312685, −1.86767772322833534227136792854, −1.79534126096451592561920406277, −1.22009592651033190998187781891, −0.61608851251157899507562863377, 0.61608851251157899507562863377, 1.22009592651033190998187781891, 1.79534126096451592561920406277, 1.86767772322833534227136792854, 2.12770916693379347348022312685, 2.62398767237310129532409966925, 3.30188167355794241111823405207, 3.40736505686345329035316448916, 3.48738172840794368963940001970, 3.56234497325956004639105698721, 4.29419513522135571983399784796, 4.65498515269625819021756396253, 4.99537457163729420102410260742, 5.25152056206810604680905359809, 5.40937285065601616465906456078, 5.44633409098404133479948268008, 5.61574650205583044526235799684, 6.02786747564235330744470976082, 6.34011920926801764563699016675, 6.57452826734288802278571971606, 6.90815536370214047351725288857, 6.99415327141256090395497325092, 7.19258741600170091374277019536, 7.68414523634273407074751041424, 7.73995901752512594892029298067

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