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 − 4·4-s + 6·9-s − 12·11-s − 16·12-s + 4·16-s − 12·17-s − 2·25-s − 4·27-s + 12·29-s + 16·31-s − 48·33-s − 24·36-s + 16·37-s − 24·41-s + 48·44-s + 16·48-s − 2·49-s − 48·51-s + 16·64-s − 16·67-s + 48·68-s − 8·75-s − 37·81-s + 12·83-s + 48·87-s + 64·93-s + ⋯
L(s)  = 1  + 2.30·3-s − 2·4-s + 2·9-s − 3.61·11-s − 4.61·12-s + 16-s − 2.91·17-s − 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 8.35·33-s − 4·36-s + 2.63·37-s − 3.74·41-s + 7.23·44-s + 2.30·48-s − 2/7·49-s − 6.72·51-s + 2·64-s − 1.95·67-s + 5.82·68-s − 0.923·75-s − 4.11·81-s + 1.31·83-s + 5.14·87-s + 6.63·93-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  =  \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.7340658765\)
\(L(\frac12)\)  \(\approx\)  \(0.7340658765\)
\(L(\frac{3}{2})\)   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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 2 p T^{2} + 1011 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 38 T^{2} + 771 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 134 T^{2} + 8115 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 16 T^{2} - 2718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 190 T^{2} + 14571 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 22 T^{2} + 3555 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 158 T^{2} + 11883 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 272 T^{2} + 30690 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 98 T^{2} + 16443 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
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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

−6.91203704590542942370675355705, −6.73041019965430853882589890832, −6.68341426209116621916871477674, −6.52321052831688718640739036650, −6.19853898811194708671325459513, −5.66180039340876746857808495419, −5.65339590942926893933508536536, −5.22339212245830237446049623334, −5.00448469238100960787864606433, −4.91179356007377517280699769106, −4.77517801951387177258266818107, −4.33492508349302071643485766775, −4.19384811111398399880030679746, −4.06983312236944761954448656367, −3.99769320202935130891428941274, −3.13364371502824762233945121972, −3.00108665063701512194098575358, −2.95791552281750598186005063717, −2.67971866236220210564290020060, −2.49604253961629437896164921632, −2.28351808249928811457742565149, −1.86924436097575112259862729840, −1.45682568409534002257902270674, −0.51968838239840333391418286012, −0.26149358816505158561078031671, 0.26149358816505158561078031671, 0.51968838239840333391418286012, 1.45682568409534002257902270674, 1.86924436097575112259862729840, 2.28351808249928811457742565149, 2.49604253961629437896164921632, 2.67971866236220210564290020060, 2.95791552281750598186005063717, 3.00108665063701512194098575358, 3.13364371502824762233945121972, 3.99769320202935130891428941274, 4.06983312236944761954448656367, 4.19384811111398399880030679746, 4.33492508349302071643485766775, 4.77517801951387177258266818107, 4.91179356007377517280699769106, 5.00448469238100960787864606433, 5.22339212245830237446049623334, 5.65339590942926893933508536536, 5.66180039340876746857808495419, 6.19853898811194708671325459513, 6.52321052831688718640739036650, 6.68341426209116621916871477674, 6.73041019965430853882589890832, 6.91203704590542942370675355705

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