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

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s + 32·9-s + 32·13-s − 17·16-s − 40·19-s − 48·25-s + 72·27-s + 120·37-s + 256·39-s + 144·43-s − 136·48-s − 98·49-s − 320·57-s + 128·67-s + 156·73-s − 384·75-s + 47·81-s + 452·97-s − 20·103-s + 960·111-s + 1.02e3·117-s + 288·121-s + 127-s + 1.15e3·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 8/3·3-s + 32/9·9-s + 2.46·13-s − 1.06·16-s − 2.10·19-s − 1.91·25-s + 8/3·27-s + 3.24·37-s + 6.56·39-s + 3.34·43-s − 2.83·48-s − 2·49-s − 5.61·57-s + 1.91·67-s + 2.13·73-s − 5.11·75-s + 0.580·81-s + 4.65·97-s − 0.194·103-s + 8.64·111-s + 8.75·117-s + 2.38·121-s + 0.00787·127-s + 8.93·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-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  =  \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(5.95063\)
\(L(\frac12)\)  \(\approx\)  \(5.95063\)
\(L(2)\)   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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
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$C_2^2$ \( ( 1 - 226 T + 25538 T^{2} - 226 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.597015855697787738483434200141, −9.543373303150697688468989652902, −9.091765553966118946111961630164, −9.089146681742194004043493140220, −8.908446731241995236782187308874, −8.367731524258278712629340586843, −8.193493020369552392437613479497, −7.990704370693596228110361884251, −7.82499744274678226631490166019, −7.51340445053640932574810495743, −7.05945642683123586463011996937, −6.48996326215910730911536666978, −6.18924078400127978636880539657, −6.18178216206250038301597745392, −5.85132540180852832653601515936, −5.06184352957247083015334092420, −4.54679444261208009815435983861, −4.11642442026930395648732055817, −3.98132795514445681081630975342, −3.68280546313233073517983183157, −3.24997244599020296648130090940, −2.55924115136985596871101929537, −2.19475981152316429990697946482, −2.08613333671172545868103919682, −1.00988996578530781446821528905, 1.00988996578530781446821528905, 2.08613333671172545868103919682, 2.19475981152316429990697946482, 2.55924115136985596871101929537, 3.24997244599020296648130090940, 3.68280546313233073517983183157, 3.98132795514445681081630975342, 4.11642442026930395648732055817, 4.54679444261208009815435983861, 5.06184352957247083015334092420, 5.85132540180852832653601515936, 6.18178216206250038301597745392, 6.18924078400127978636880539657, 6.48996326215910730911536666978, 7.05945642683123586463011996937, 7.51340445053640932574810495743, 7.82499744274678226631490166019, 7.990704370693596228110361884251, 8.193493020369552392437613479497, 8.367731524258278712629340586843, 8.908446731241995236782187308874, 9.089146681742194004043493140220, 9.091765553966118946111961630164, 9.543373303150697688468989652902, 9.597015855697787738483434200141

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