Properties

Degree 8
Conductor $ 3^{8} \cdot 43^{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  + 6·2-s + 17·4-s − 2·5-s + 4·7-s + 30·8-s − 12·10-s + 10·11-s − 5·13-s + 24·14-s + 40·16-s + 17-s − 2·19-s − 34·20-s + 60·22-s + 11·23-s + 6·25-s − 30·26-s + 68·28-s + 6·29-s + 54·32-s + 6·34-s − 8·35-s + 3·37-s − 12·38-s − 60·40-s − 20·41-s − 26·43-s + ⋯
L(s)  = 1  + 4.24·2-s + 17/2·4-s − 0.894·5-s + 1.51·7-s + 10.6·8-s − 3.79·10-s + 3.01·11-s − 1.38·13-s + 6.41·14-s + 10·16-s + 0.242·17-s − 0.458·19-s − 7.60·20-s + 12.7·22-s + 2.29·23-s + 6/5·25-s − 5.88·26-s + 12.8·28-s + 1.11·29-s + 9.54·32-s + 1.02·34-s − 1.35·35-s + 0.493·37-s − 1.94·38-s − 9.48·40-s − 3.12·41-s − 3.96·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 43^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{387} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{8} \cdot 43^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(26.2879\)
\(L(\frac12)\)  \(\approx\)  \(26.2879\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;43\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 8 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 4 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 25 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - T - 2 T^{2} + 31 T^{3} - 297 T^{4} + 31 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 8 p T^{5} - 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 11 T + 2 p T^{2} - 319 T^{3} + 2313 T^{4} - 319 p T^{5} + 2 p^{3} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 27 p T^{5} - 56 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 5 T - 86 T^{2} + 25 T^{3} + 8187 T^{4} + 25 p T^{5} - 86 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 11 p T^{5} - 110 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - T - 122 T^{2} + 11 T^{3} + 10573 T^{4} + 11 p T^{5} - 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 447 p T^{5} + 16 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 27 p T^{5} - 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 5 T - 138 T^{2} - 25 T^{3} + 18353 T^{4} - 25 p T^{5} - 138 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 627 p T^{5} + 52 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 3 T - 160 T^{2} + 27 T^{3} + 19839 T^{4} + 27 p T^{5} - 160 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.216962017218923593468545866903, −8.016124011734704576909192541861, −7.46024258965996841981214748531, −7.14967501520449008266804408669, −6.93512064305874472716357360859, −6.77415307266928779591835410604, −6.58915341542623029717931523263, −6.31475717651506146810925655415, −6.24092857516818876140666146577, −5.39760987428249841793093406373, −5.38609937878047964828311711845, −5.16876031625726942726943532333, −5.02367580718238670301237165275, −4.69882702522476547129032413739, −4.43861141019056810970362673115, −4.36408407197613184849446283523, −4.33339917858814747828843287639, −3.53333195714306311824259795575, −3.33167041264495287755344512937, −3.31863811460991859938183471188, −3.31530015010412508927673336024, −2.56327692664283757678581075940, −1.85514563682022815991985534519, −1.55474789094885735567747694563, −1.16138409765984000592841574666, 1.16138409765984000592841574666, 1.55474789094885735567747694563, 1.85514563682022815991985534519, 2.56327692664283757678581075940, 3.31530015010412508927673336024, 3.31863811460991859938183471188, 3.33167041264495287755344512937, 3.53333195714306311824259795575, 4.33339917858814747828843287639, 4.36408407197613184849446283523, 4.43861141019056810970362673115, 4.69882702522476547129032413739, 5.02367580718238670301237165275, 5.16876031625726942726943532333, 5.38609937878047964828311711845, 5.39760987428249841793093406373, 6.24092857516818876140666146577, 6.31475717651506146810925655415, 6.58915341542623029717931523263, 6.77415307266928779591835410604, 6.93512064305874472716357360859, 7.14967501520449008266804408669, 7.46024258965996841981214748531, 8.016124011734704576909192541861, 8.216962017218923593468545866903

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