Properties

Degree 8
Conductor $ 2^{8} \cdot 31^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯
L(s)  = 1  − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 31^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{124} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 31^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.07615071625$
$L(\frac12)$  $\approx$  $0.07615071625$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;31\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;31\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
79$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−10.25696943349034234137116405243, −9.935290020569122213739987438421, −9.326942655554410349488623302761, −9.265677000862623320034465975496, −8.768357939163126364910945396664, −8.622440126044797827118706350253, −8.536369759493712591970660613850, −8.387361065421477009127377174460, −8.302147952705361882193863854075, −7.55840518286483759960699494642, −7.35518884853854989998619678137, −7.21735457249255791734349745664, −6.51577319627012882774980448639, −6.30474776983992920335379169545, −6.18483581618578829344299054578, −5.41607412047591697559067547569, −5.15770407288810986410585864163, −5.01178891049608934019955746078, −4.55972529911887285331881387887, −4.01588990366557451481704433847, −3.94693333798037946909518849036, −3.62548533295285780113509540828, −3.23664863338373525323008173673, −2.76326840611427196538532991642, −1.52908883261400833847866302720, 1.52908883261400833847866302720, 2.76326840611427196538532991642, 3.23664863338373525323008173673, 3.62548533295285780113509540828, 3.94693333798037946909518849036, 4.01588990366557451481704433847, 4.55972529911887285331881387887, 5.01178891049608934019955746078, 5.15770407288810986410585864163, 5.41607412047591697559067547569, 6.18483581618578829344299054578, 6.30474776983992920335379169545, 6.51577319627012882774980448639, 7.21735457249255791734349745664, 7.35518884853854989998619678137, 7.55840518286483759960699494642, 8.302147952705361882193863854075, 8.387361065421477009127377174460, 8.536369759493712591970660613850, 8.622440126044797827118706350253, 8.768357939163126364910945396664, 9.265677000862623320034465975496, 9.326942655554410349488623302761, 9.935290020569122213739987438421, 10.25696943349034234137116405243

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