Properties

Degree 8
Conductor $ 3^{8} \cdot 7^{4} \cdot 11^{8} $
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  − 2-s − 2·4-s + 4·5-s − 4·7-s + 6·8-s − 4·10-s + 6·13-s + 4·14-s − 4·16-s − 8·17-s − 10·19-s − 8·20-s + 10·23-s + 4·25-s − 6·26-s + 8·28-s − 18·31-s − 11·32-s + 8·34-s − 16·35-s − 2·37-s + 10·38-s + 24·40-s − 10·41-s − 4·43-s − 10·46-s − 4·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s + 1.78·5-s − 1.51·7-s + 2.12·8-s − 1.26·10-s + 1.66·13-s + 1.06·14-s − 16-s − 1.94·17-s − 2.29·19-s − 1.78·20-s + 2.08·23-s + 4/5·25-s − 1.17·26-s + 1.51·28-s − 3.23·31-s − 1.94·32-s + 1.37·34-s − 2.70·35-s − 0.328·37-s + 1.62·38-s + 3.79·40-s − 1.56·41-s − 0.609·43-s − 1.47·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $0.4349444950$
$L(\frac12)$  $\approx$  $0.4349444950$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 3 T^{2} - T^{3} + 3 T^{4} - p T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 44 T^{2} - 218 T^{3} + 822 T^{4} - 218 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 48 T^{2} + 264 T^{3} + 1358 T^{4} + 264 p T^{5} + 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 100 T^{2} + 30 p T^{3} + 3062 T^{4} + 30 p^{2} T^{5} + 100 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 83 T^{2} - 500 T^{3} + 2869 T^{4} - 500 p T^{5} + 83 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + 37 T^{2} + 17 p T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 1362 p T^{5} + 192 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 4 p T^{5} + 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 60 T^{2} - 290 T^{3} - 2938 T^{4} - 290 p T^{5} + 60 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 392 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 108 T^{2} - 4 T^{3} + 4758 T^{4} - 4 p T^{5} + 108 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 161 T^{2} + 20 T^{3} + 11657 T^{4} + 20 p T^{5} + 161 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 308 T^{2} - 2936 T^{3} + 29398 T^{4} - 2936 p T^{5} + 308 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 260 T^{2} - 2306 T^{3} + 24294 T^{4} - 2306 p T^{5} + 260 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 6696 p T^{5} + 541 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 327 T^{2} - 3632 T^{3} + 36173 T^{4} - 3632 p T^{5} + 327 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 284 T^{2} + 852 T^{3} + 30822 T^{4} + 852 p T^{5} + 284 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 445 T^{2} + 5040 T^{3} + 57977 T^{4} + 5040 p T^{5} + 445 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 212 T^{2} + 982 T^{3} + 24278 T^{4} + 982 p T^{5} + 212 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 308 T^{2} - 3096 T^{3} + 38678 T^{4} - 3096 p T^{5} + 308 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 8944 p T^{5} + 656 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.74483373697436523159097395089, −5.22168789841786605527993598637, −5.09039809734285007109952613905, −4.98145696792792449552037861277, −4.91984970702585822060813618190, −4.63979131960618953964648500112, −4.30620291669772742240020544636, −4.27440674080925693175285995435, −4.12693570707946889029657033695, −3.64554719061296354498366202123, −3.56959499915042015431233962641, −3.51007879565810396693399221280, −3.49279200686649778081064411031, −3.03461197239554307608136464701, −2.62226773967796975621425380241, −2.53357300771486945046007414706, −2.19599488161042912285673705934, −2.03517294734479608451726986795, −1.82410748104880055423200393360, −1.80400960759540608039225370140, −1.42677117157122029804410031917, −1.21683357864574674620798060501, −0.74772146701630294207223469872, −0.44435449608172592586083596473, −0.13118864482230380891724525022, 0.13118864482230380891724525022, 0.44435449608172592586083596473, 0.74772146701630294207223469872, 1.21683357864574674620798060501, 1.42677117157122029804410031917, 1.80400960759540608039225370140, 1.82410748104880055423200393360, 2.03517294734479608451726986795, 2.19599488161042912285673705934, 2.53357300771486945046007414706, 2.62226773967796975621425380241, 3.03461197239554307608136464701, 3.49279200686649778081064411031, 3.51007879565810396693399221280, 3.56959499915042015431233962641, 3.64554719061296354498366202123, 4.12693570707946889029657033695, 4.27440674080925693175285995435, 4.30620291669772742240020544636, 4.63979131960618953964648500112, 4.91984970702585822060813618190, 4.98145696792792449552037861277, 5.09039809734285007109952613905, 5.22168789841786605527993598637, 5.74483373697436523159097395089

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