Properties

Label 8-7200e4-1.1-c1e4-0-36
Degree $8$
Conductor $2.687\times 10^{15}$
Sign $1$
Analytic cond. $1.09254\times 10^{7}$
Root an. cond. $7.58236$
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  + 24·29-s − 8·41-s + 12·49-s − 8·61-s + 40·89-s + 8·101-s + 72·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4.45·29-s − 1.24·41-s + 12/7·49-s − 1.02·61-s + 4.23·89-s + 0.796·101-s + 6.89·109-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.09254\times 10^{7}\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.406894483\)
\(L(\frac12)\) \(\approx\) \(9.406894483\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.61633048900528609860934201220, −5.19218545621568710898378051732, −5.15305494852376585102118142553, −5.01141070697186690976656219957, −4.83531634803333086873164805530, −4.57668138792436507615124268942, −4.42741845741385986054532101570, −4.42686598211880121812867009323, −4.20340810293441425257150978015, −3.70939780375753892015849306796, −3.57835621788388802753566999296, −3.45821674675477819568040917461, −3.35003215205336487942276309473, −2.95501577763324480433586879451, −2.79012952573848362269297956972, −2.69922534532862369397671722693, −2.50122322779165823029852736069, −2.04335727876463045059910511894, −1.87085214321009233565149092071, −1.82703938522790392996229776317, −1.49790607187681915886781130889, −0.879592843478337469356589750622, −0.812763391941497603436675494074, −0.72303286314502468889046315025, −0.41996540591292181751682563827, 0.41996540591292181751682563827, 0.72303286314502468889046315025, 0.812763391941497603436675494074, 0.879592843478337469356589750622, 1.49790607187681915886781130889, 1.82703938522790392996229776317, 1.87085214321009233565149092071, 2.04335727876463045059910511894, 2.50122322779165823029852736069, 2.69922534532862369397671722693, 2.79012952573848362269297956972, 2.95501577763324480433586879451, 3.35003215205336487942276309473, 3.45821674675477819568040917461, 3.57835621788388802753566999296, 3.70939780375753892015849306796, 4.20340810293441425257150978015, 4.42686598211880121812867009323, 4.42741845741385986054532101570, 4.57668138792436507615124268942, 4.83531634803333086873164805530, 5.01141070697186690976656219957, 5.15305494852376585102118142553, 5.19218545621568710898378051732, 5.61633048900528609860934201220

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