Properties

Degree 8
Conductor $ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{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  + 4·3-s − 4-s − 4·5-s − 4·7-s + 10·9-s − 4·11-s − 4·12-s + 8·13-s − 16·15-s + 3·16-s − 2·17-s + 10·19-s + 4·20-s − 16·21-s − 2·23-s + 10·25-s + 20·27-s + 4·28-s − 2·29-s + 24·31-s − 16·33-s + 16·35-s − 10·36-s + 8·37-s + 32·39-s + 6·43-s + 4·44-s + ⋯
L(s)  = 1  + 2.30·3-s − 1/2·4-s − 1.78·5-s − 1.51·7-s + 10/3·9-s − 1.20·11-s − 1.15·12-s + 2.21·13-s − 4.13·15-s + 3/4·16-s − 0.485·17-s + 2.29·19-s + 0.894·20-s − 3.49·21-s − 0.417·23-s + 2·25-s + 3.84·27-s + 0.755·28-s − 0.371·29-s + 4.31·31-s − 2.78·33-s + 2.70·35-s − 5/3·36-s + 1.31·37-s + 5.12·39-s + 0.914·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1155} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(7.297099512\)
\(L(\frac12)\)  \(\approx\)  \(7.297099512\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$C_2^2 \wr C_2$ \( 1 + T^{2} - p T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 50 T^{2} - 240 T^{3} + 1002 T^{4} - 240 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 37 T^{2} + 78 T^{3} - 916 T^{4} + 78 p T^{5} + 37 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 21 T^{2} - 98 T^{3} - 108 T^{4} - 98 p T^{5} + 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 71 T^{2} + 210 T^{3} + 2432 T^{4} + 210 p T^{5} + 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 326 T^{2} - 2928 T^{3} + 19090 T^{4} - 2928 p T^{5} + 326 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 114 T^{2} - 688 T^{3} + 6282 T^{4} - 688 p T^{5} + 114 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2 \wr C_2$ \( 1 + 38 T^{2} + 402 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 107 T^{2} - 470 T^{3} + 5520 T^{4} - 470 p T^{5} + 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 54 T^{2} - 452 T^{3} + 786 T^{4} - 452 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 161 T^{2} - 1198 T^{3} + 10012 T^{4} - 1198 p T^{5} + 161 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 203 T^{2} + 238 T^{3} + 16912 T^{4} + 238 p T^{5} + 203 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 121 T^{2} - 866 T^{3} + 9380 T^{4} - 866 p T^{5} + 121 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 176 T^{2} + 1176 T^{3} + 17358 T^{4} + 1176 p T^{5} + 176 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 + 258 T^{2} + 26682 T^{4} + 258 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 112 T^{2} + 148 T^{3} + 4318 T^{4} + 148 p T^{5} + 112 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 + 290 T^{2} + 33466 T^{4} + 290 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 117 T^{2} - 1478 T^{3} + 148 p T^{4} - 1478 p T^{5} + 117 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 255 T^{2} - 2594 T^{3} + 29096 T^{4} - 2594 p T^{5} + 255 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 111 T^{2} - 834 T^{3} - 984 T^{4} - 834 p T^{5} + 111 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.07230571138125899395847677134, −6.99124926173308188664323094612, −6.50381514902619309030818384640, −6.48237852303071604337493604741, −6.21146211996191133827601478262, −5.99669726581949430975170155912, −5.58597614479635936194108424768, −5.49785674740919231189777543041, −5.19062424828374817329144499381, −4.71013548150215139994112787272, −4.47224355619173964652126025914, −4.35541577843346628666344730143, −4.17388398336443503037997852654, −3.88965084029950774652651027603, −3.65477393895859259061618253869, −3.36591280884191110978989785576, −3.03622844816130731329806270061, −3.03120596655070389442266982397, −3.02548898860575796164904232976, −2.46595098435056589530916554341, −2.26836298564414965110307697230, −1.73395395791610990103311187312, −0.955862374071830574861273941973, −0.854303664422338879810808974059, −0.76246366574970015371563127252, 0.76246366574970015371563127252, 0.854303664422338879810808974059, 0.955862374071830574861273941973, 1.73395395791610990103311187312, 2.26836298564414965110307697230, 2.46595098435056589530916554341, 3.02548898860575796164904232976, 3.03120596655070389442266982397, 3.03622844816130731329806270061, 3.36591280884191110978989785576, 3.65477393895859259061618253869, 3.88965084029950774652651027603, 4.17388398336443503037997852654, 4.35541577843346628666344730143, 4.47224355619173964652126025914, 4.71013548150215139994112787272, 5.19062424828374817329144499381, 5.49785674740919231189777543041, 5.58597614479635936194108424768, 5.99669726581949430975170155912, 6.21146211996191133827601478262, 6.48237852303071604337493604741, 6.50381514902619309030818384640, 6.99124926173308188664323094612, 7.07230571138125899395847677134

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