Properties

Degree 8
Conductor $ 43^{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  − 6·2-s + 3·3-s + 17·4-s + 2·5-s − 18·6-s + 4·7-s − 30·8-s + 7·9-s − 12·10-s − 10·11-s + 51·12-s − 5·13-s − 24·14-s + 6·15-s + 40·16-s − 17-s − 42·18-s − 2·19-s + 34·20-s + 12·21-s + 60·22-s − 11·23-s − 90·24-s + 6·25-s + 30·26-s + 18·27-s + 68·28-s + ⋯
L(s)  = 1  − 4.24·2-s + 1.73·3-s + 17/2·4-s + 0.894·5-s − 7.34·6-s + 1.51·7-s − 10.6·8-s + 7/3·9-s − 3.79·10-s − 3.01·11-s + 14.7·12-s − 1.38·13-s − 6.41·14-s + 1.54·15-s + 10·16-s − 0.242·17-s − 9.89·18-s − 0.458·19-s + 7.60·20-s + 2.61·21-s + 12.7·22-s − 2.29·23-s − 18.3·24-s + 6/5·25-s + 5.88·26-s + 3.46·27-s + 12.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3418801\)    =    \(43^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3418801,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.0986474\)
\(L(\frac12)\)  \(\approx\)  \(0.0986474\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 8. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
3$D_4\times C_2$ \( 1 - p T + 2 T^{2} - p T^{3} + 13 T^{4} - p^{2} T^{5} + 2 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
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

−12.06634139006083615744554081716, −11.34106048485691358533374741013, −10.75863954890462280061334070438, −10.72338412949536074580214522873, −10.45337744844734479329383752623, −10.10430762237744544771690344315, −9.922625935869659530103646991449, −9.790324727253329212579557074059, −9.349533236810762881900709650631, −9.065498323329843765523844033472, −8.685150939867275425868089843298, −8.423380481787101835574062026676, −8.260519378923592803666677811258, −7.926082290983334950019780376527, −7.65308196408449499798381789852, −7.44950380377139430208752767735, −7.30586229910336100940751206951, −6.46904971033821410527799815476, −5.66989695649487317765268894238, −5.31054274513364251844579521823, −4.74613885977079331036065932448, −4.06187580240804674038035488591, −2.55953159545145440458288446491, −2.54138790800309779570144036565, −1.74402716981569586677513667000, 1.74402716981569586677513667000, 2.54138790800309779570144036565, 2.55953159545145440458288446491, 4.06187580240804674038035488591, 4.74613885977079331036065932448, 5.31054274513364251844579521823, 5.66989695649487317765268894238, 6.46904971033821410527799815476, 7.30586229910336100940751206951, 7.44950380377139430208752767735, 7.65308196408449499798381789852, 7.926082290983334950019780376527, 8.260519378923592803666677811258, 8.423380481787101835574062026676, 8.685150939867275425868089843298, 9.065498323329843765523844033472, 9.349533236810762881900709650631, 9.790324727253329212579557074059, 9.922625935869659530103646991449, 10.10430762237744544771690344315, 10.45337744844734479329383752623, 10.72338412949536074580214522873, 10.75863954890462280061334070438, 11.34106048485691358533374741013, 12.06634139006083615744554081716

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