Properties

Degree 8
Conductor $ 3^{4} \cdot 5^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·3-s − 28·7-s + 32·9-s − 32·13-s − 17·16-s + 40·19-s + 224·21-s − 48·25-s − 72·27-s + 120·37-s + 256·39-s + 144·43-s + 136·48-s + 490·49-s − 320·57-s − 896·63-s + 128·67-s − 156·73-s + 384·75-s + 47·81-s + 896·91-s − 452·97-s + 20·103-s − 960·111-s + 476·112-s − 1.02e3·117-s + 288·121-s + ⋯
L(s)  = 1  − 8/3·3-s − 4·7-s + 32/9·9-s − 2.46·13-s − 1.06·16-s + 2.10·19-s + 32/3·21-s − 1.91·25-s − 8/3·27-s + 3.24·37-s + 6.56·39-s + 3.34·43-s + 17/6·48-s + 10·49-s − 5.61·57-s − 14.2·63-s + 1.91·67-s − 2.13·73-s + 5.11·75-s + 0.580·81-s + 9.84·91-s − 4.65·97-s + 0.194·103-s − 8.64·111-s + 17/4·112-s − 8.75·117-s + 2.38·121-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.183485\)
\(L(\frac12)\)  \(\approx\)  \(0.183485\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 48 T^{2} + p^{4} T^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2^3$ \( 1 + 17 T^{4} + p^{8} T^{8} \)
11$C_2^2$ \( ( 1 - 144 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 157438 T^{4} + p^{8} T^{8} \)
19$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{4} \)
23$C_2^3$ \( 1 - 550078 T^{4} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 + 1664 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1726 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 60 T + 1800 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 2210 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 9442562 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 3420962 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 - 6080 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 7246 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 64 T + 2048 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 6554 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 9346 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 27716834 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 15450 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 226 T + 25538 T^{2} + 226 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−9.811745863501089222348002304897, −9.750950037376473135474802278919, −9.547207636078389798051550982679, −9.477284155170749861766325078059, −9.112816842924075043703766906012, −8.643648022220908877723773114105, −7.85338330636812864145274429668, −7.53534504763724249964030553505, −7.37615866519857267870394514146, −7.06919363656686761701684223736, −6.85351808989082253946203402855, −6.52546694774950296529132460344, −6.01383823634531880522213980619, −5.97730715584268499076089613183, −5.84663728465937851000850026781, −5.32538334065141875136156967179, −5.18706757672327632885740187913, −4.41699730211243890148507602821, −4.08777774079155002828922460730, −3.97895784356392469049813223415, −3.02843516296389045995324115271, −2.62513825992196627379721604839, −2.60037500277158521014176848205, −0.66972576015080695041322104618, −0.39930638394500475769208940704, 0.39930638394500475769208940704, 0.66972576015080695041322104618, 2.60037500277158521014176848205, 2.62513825992196627379721604839, 3.02843516296389045995324115271, 3.97895784356392469049813223415, 4.08777774079155002828922460730, 4.41699730211243890148507602821, 5.18706757672327632885740187913, 5.32538334065141875136156967179, 5.84663728465937851000850026781, 5.97730715584268499076089613183, 6.01383823634531880522213980619, 6.52546694774950296529132460344, 6.85351808989082253946203402855, 7.06919363656686761701684223736, 7.37615866519857267870394514146, 7.53534504763724249964030553505, 7.85338330636812864145274429668, 8.643648022220908877723773114105, 9.112816842924075043703766906012, 9.477284155170749861766325078059, 9.547207636078389798051550982679, 9.750950037376473135474802278919, 9.811745863501089222348002304897

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