Properties

Label 8-475e4-1.1-c1e4-0-1
Degree $8$
Conductor $50906640625$
Sign $1$
Analytic cond. $206.958$
Root an. cond. $1.94753$
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  − 6·7-s − 8·16-s + 14·17-s − 8·23-s + 2·43-s − 26·47-s + 18·49-s + 22·73-s − 18·81-s + 32·83-s + 48·112-s − 84·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2.26·7-s − 2·16-s + 3.39·17-s − 1.66·23-s + 0.304·43-s − 3.79·47-s + 18/7·49-s + 2.57·73-s − 2·81-s + 3.51·83-s + 4.53·112-s − 7.70·119-s − 0.545·121-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 + 3.78·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(206.958\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6492179306\)
\(L(\frac12)\) \(\approx\) \(0.6492179306\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2$$\times$$C_2^2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 15 T^{2} + p^{2} T^{4} ) \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 30 T^{2} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 - 85 T^{2} + p^{2} T^{4} ) \)
47$C_2$$\times$$C_2^2$ \( ( 1 + 13 T + p T^{2} )^{2}( 1 + 75 T^{2} + p^{2} T^{4} ) \)
53$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2$$\times$$C_2^2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2$$\times$$C_2^2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 90 T^{2} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
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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

−7.84509330931984383084080124554, −7.70708095785568333377653997360, −7.66273146773971366844500387611, −7.11535433642295865400163367946, −7.01987432224481562987139020813, −6.51591101344694365055661929044, −6.44941810587166817618596429198, −6.37954543982250408774241704807, −6.17013184146683492010185497715, −5.67285788013934757179832067609, −5.58508075548018664640411316183, −5.10814855300203410242461842413, −5.05351519780631410895771827121, −4.70778975654487276188941393264, −4.29287209788908344502150817662, −3.79881751746183031268491832202, −3.67502021899078383462489706737, −3.56657949847332998137500340845, −3.10852318202880776455356011934, −2.94521458717871129049045391629, −2.51759522165962650799529371831, −2.09818360877445881677812392802, −1.68249617262429988445591308594, −1.06723956875773738720441433594, −0.30374544067046084425580504143, 0.30374544067046084425580504143, 1.06723956875773738720441433594, 1.68249617262429988445591308594, 2.09818360877445881677812392802, 2.51759522165962650799529371831, 2.94521458717871129049045391629, 3.10852318202880776455356011934, 3.56657949847332998137500340845, 3.67502021899078383462489706737, 3.79881751746183031268491832202, 4.29287209788908344502150817662, 4.70778975654487276188941393264, 5.05351519780631410895771827121, 5.10814855300203410242461842413, 5.58508075548018664640411316183, 5.67285788013934757179832067609, 6.17013184146683492010185497715, 6.37954543982250408774241704807, 6.44941810587166817618596429198, 6.51591101344694365055661929044, 7.01987432224481562987139020813, 7.11535433642295865400163367946, 7.66273146773971366844500387611, 7.70708095785568333377653997360, 7.84509330931984383084080124554

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