Properties

Label 8-585e4-1.1-c1e4-0-9
Degree $8$
Conductor $117117950625$
Sign $1$
Analytic cond. $476.136$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 21·9-s + 6·13-s + 4·16-s − 4·17-s − 2·23-s + 25-s + 54·27-s − 4·29-s + 36·39-s − 10·43-s + 24·48-s − 10·49-s − 24·51-s + 36·53-s + 10·61-s − 12·69-s + 6·75-s − 2·79-s + 108·81-s − 24·87-s − 14·101-s − 32·103-s + 68·107-s − 38·113-s + 126·117-s + 14·121-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s + 1.66·13-s + 16-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 10.3·27-s − 0.742·29-s + 5.76·39-s − 1.52·43-s + 3.46·48-s − 1.42·49-s − 3.36·51-s + 4.94·53-s + 1.28·61-s − 1.44·69-s + 0.692·75-s − 0.225·79-s + 12·81-s − 2.57·87-s − 1.39·101-s − 3.15·103-s + 6.57·107-s − 3.57·113-s + 11.6·117-s + 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(476.136\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.70439053\)
\(L(\frac12)\) \(\approx\) \(13.70439053\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \)
7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^3$ \( 1 - 2 T^{2} - 957 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^3$ \( 1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
47$C_2^3$ \( 1 - 6 T^{2} - 2173 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 + 82 T^{2} + 3243 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2^3$ \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.938597636933037529161568732985, −7.79301199580912705483573447502, −7.22837849844471953703003316850, −7.09454021722035458767763847018, −6.91133765759704696590064958901, −6.66431488351843470087323458027, −6.58438856348517467870276561944, −5.97648769695529191017943049092, −5.86129372743986573456806269248, −5.40162616508879279770677284158, −5.35075070604559398945654636208, −4.95233694432358648576112464374, −4.36964616414684454371753312904, −4.19492163564288240336668589342, −3.99733983962288186047700822684, −3.96799801616899656359703122784, −3.37431005155124049813486343894, −3.34617826599026920783086675271, −3.10757297454968280424412878218, −2.74675631056328858719227453352, −2.18566150971684533201942943300, −2.16296743276808025450618966523, −1.83309061598921064154537107486, −1.31908878548689850259178556920, −0.923451805834034125913335803134, 0.923451805834034125913335803134, 1.31908878548689850259178556920, 1.83309061598921064154537107486, 2.16296743276808025450618966523, 2.18566150971684533201942943300, 2.74675631056328858719227453352, 3.10757297454968280424412878218, 3.34617826599026920783086675271, 3.37431005155124049813486343894, 3.96799801616899656359703122784, 3.99733983962288186047700822684, 4.19492163564288240336668589342, 4.36964616414684454371753312904, 4.95233694432358648576112464374, 5.35075070604559398945654636208, 5.40162616508879279770677284158, 5.86129372743986573456806269248, 5.97648769695529191017943049092, 6.58438856348517467870276561944, 6.66431488351843470087323458027, 6.91133765759704696590064958901, 7.09454021722035458767763847018, 7.22837849844471953703003316850, 7.79301199580912705483573447502, 7.938597636933037529161568732985

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