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  − 2·2-s + 6·5-s − 4·7-s + 8-s − 12·10-s + 8·14-s − 16-s + 3·17-s + 3·19-s + 8·23-s + 8·25-s − 3·29-s − 3·31-s + 10·32-s − 6·34-s − 24·35-s − 7·37-s − 6·38-s + 6·40-s − 4·41-s + 8·43-s − 16·46-s + 14·47-s + 10·49-s − 16·50-s + 9·53-s − 4·56-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.68·5-s − 1.51·7-s + 0.353·8-s − 3.79·10-s + 2.13·14-s − 1/4·16-s + 0.727·17-s + 0.688·19-s + 1.66·23-s + 8/5·25-s − 0.557·29-s − 0.538·31-s + 1.76·32-s − 1.02·34-s − 4.05·35-s − 1.15·37-s − 0.973·38-s + 0.948·40-s − 0.624·41-s + 1.21·43-s − 2.35·46-s + 2.04·47-s + 10/7·49-s − 2.26·50-s + 1.23·53-s − 0.534·56-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  =  $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $4.629175627$
$L(\frac12)$  $\approx$  $4.629175627$
$L(\frac{3}{2})$   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$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + 7 T^{3} + 13 T^{4} + 7 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 87 T^{3} + 229 T^{4} - 87 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} - 5 p T^{3} + 153 T^{4} - 5 p^{2} T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 45 T^{2} - 62 T^{3} + 881 T^{4} - 62 p T^{5} + 45 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 47 T^{2} - 36 T^{3} + 919 T^{4} - 36 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 402 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 62 T^{2} + 9 p T^{3} + 2319 T^{4} + 9 p^{2} T^{5} + 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 100 T^{2} + 299 T^{3} + 4283 T^{4} + 299 p T^{5} + 100 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 767 p T^{5} + 154 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 107 T^{2} + 470 T^{3} + 5491 T^{4} + 470 p T^{5} + 107 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 162 T^{2} - 1449 T^{3} + 10505 T^{4} - 1449 p T^{5} + 162 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 108 T^{2} - 725 T^{3} + 4961 T^{4} - 725 p T^{5} + 108 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 25 T + 428 T^{2} - 4725 T^{3} + 42353 T^{4} - 4725 p T^{5} + 428 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 238 T^{2} + 2447 T^{3} + 20599 T^{4} + 2447 p T^{5} + 238 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 201 T^{2} - 812 T^{3} + 17469 T^{4} - 812 p T^{5} + 201 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 176 T^{2} + 1649 T^{3} + 20013 T^{4} + 1649 p T^{5} + 176 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 308 T^{2} - 1735 T^{3} + 35911 T^{4} - 1735 p T^{5} + 308 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - T + 96 T^{2} - 1149 T^{3} + 4403 T^{4} - 1149 p T^{5} + 96 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 3419 p T^{5} + 312 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 278 T^{2} + 2415 T^{3} + 30889 T^{4} + 2415 p T^{5} + 278 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.64959413633457286904005961081, −5.35992946914113371590051322724, −5.34513795999785343226629224695, −5.20349060346069109342414548661, −4.94910520429000959012218499147, −4.37995559724002057813364698914, −4.36925818392980482824283991653, −4.31999238814034425015889028604, −4.22668352078693290504276487353, −3.69172126476643500723740512163, −3.37524342248790485438630210689, −3.32238354677886329507398542444, −3.27313137801479796917678893332, −2.91039452501965914964152793777, −2.89272923623505996682177100034, −2.42360475290526646485775332060, −2.21789584861546931770932780590, −2.12892942546241648956504482384, −1.78019484985095762085578435712, −1.69661639267176711428418975807, −1.57633295442286144064261307414, −0.72496950403575526330106149715, −0.71916198273118171598871016749, −0.57431656039733675337272187797, −0.57279222918105059945122563427, 0.57279222918105059945122563427, 0.57431656039733675337272187797, 0.71916198273118171598871016749, 0.72496950403575526330106149715, 1.57633295442286144064261307414, 1.69661639267176711428418975807, 1.78019484985095762085578435712, 2.12892942546241648956504482384, 2.21789584861546931770932780590, 2.42360475290526646485775332060, 2.89272923623505996682177100034, 2.91039452501965914964152793777, 3.27313137801479796917678893332, 3.32238354677886329507398542444, 3.37524342248790485438630210689, 3.69172126476643500723740512163, 4.22668352078693290504276487353, 4.31999238814034425015889028604, 4.36925818392980482824283991653, 4.37995559724002057813364698914, 4.94910520429000959012218499147, 5.20349060346069109342414548661, 5.34513795999785343226629224695, 5.35992946914113371590051322724, 5.64959413633457286904005961081

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