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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 4·11-s − 8·23-s − 24·37-s + 12·43-s + 8·49-s + 4·53-s + 4·67-s + 24·71-s − 16·77-s + 17·81-s + 12·107-s + 48·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s − 1.66·23-s − 3.94·37-s + 1.82·43-s + 8/7·49-s + 0.549·53-s + 0.488·67-s + 2.84·71-s − 1.82·77-s + 17/9·81-s + 1.16·107-s + 4.51·113-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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  =  \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.944374\)
\(L(\frac12)\)  \(\approx\)  \(0.944374\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 - 17 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 103 T^{4} + p^{4} T^{8} \)
17$C_2^3$ \( 1 + 263 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 2017 T^{4} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 7538 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 16903 T^{4} + p^{4} T^{8} \)
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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

−7.72870501283941326180974429356, −7.43275282023006510434238821749, −7.34429990007041802233415763209, −6.89928301812361060258880972407, −6.80956830735188032586319270898, −6.51310654736435271553296441340, −6.42672862580717191547418087734, −6.21015018296273942893591595611, −5.75380949491179843479477291476, −5.61972507577436832496684517610, −5.50877598818272985136276309381, −4.97456812635753227234560193581, −4.70202886114552999682199541047, −4.66546324608245500627467459546, −3.94235624711033108667350660203, −3.85688045322793135192724766793, −3.58411013986060227668253818799, −3.48979161712265632384259784247, −3.31094172158265691858067997645, −2.43321540535345335677756445630, −2.39107749742439765669117329982, −2.17170321123647421157774710098, −1.49650498188424158297971662093, −1.11589356614404738373358188247, −0.32324884696239103516372347832, 0.32324884696239103516372347832, 1.11589356614404738373358188247, 1.49650498188424158297971662093, 2.17170321123647421157774710098, 2.39107749742439765669117329982, 2.43321540535345335677756445630, 3.31094172158265691858067997645, 3.48979161712265632384259784247, 3.58411013986060227668253818799, 3.85688045322793135192724766793, 3.94235624711033108667350660203, 4.66546324608245500627467459546, 4.70202886114552999682199541047, 4.97456812635753227234560193581, 5.50877598818272985136276309381, 5.61972507577436832496684517610, 5.75380949491179843479477291476, 6.21015018296273942893591595611, 6.42672862580717191547418087734, 6.51310654736435271553296441340, 6.80956830735188032586319270898, 6.89928301812361060258880972407, 7.34429990007041802233415763209, 7.43275282023006510434238821749, 7.72870501283941326180974429356

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