Properties

Label 8-1200e4-1.1-c1e4-0-23
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
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  + 4·3-s + 8·7-s + 8·9-s + 32·21-s + 12·27-s − 32·31-s + 32·37-s + 8·43-s + 32·49-s − 24·61-s + 64·63-s + 24·67-s + 32·73-s + 23·81-s − 128·93-s − 32·97-s + 24·103-s + 128·111-s − 20·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s + 128·147-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.30·3-s + 3.02·7-s + 8/3·9-s + 6.98·21-s + 2.30·27-s − 5.74·31-s + 5.26·37-s + 1.21·43-s + 32/7·49-s − 3.07·61-s + 8.06·63-s + 2.93·67-s + 3.74·73-s + 23/9·81-s − 13.2·93-s − 3.24·97-s + 2.36·103-s + 12.1·111-s − 1.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.37499344\)
\(L(\frac12)\) \(\approx\) \(15.37499344\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 254 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 958 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 3682 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 3854 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^3$ \( 1 - 12878 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.30176268630976462455590035633, −6.81067793268610892919957775478, −6.66371169786463120612432130959, −6.28702418207927132833153110643, −6.14045354495691804323772519623, −5.67939968546028666580925107230, −5.56177397386920940660470103374, −5.34461845825756923269238284504, −5.21088399746162834604688941721, −4.87869756363237281080463613951, −4.44310118815521598289723043565, −4.38935634780136278507682049183, −4.37721507507315872049635166110, −3.83289836781046076663451538097, −3.62329009957845013216448332342, −3.54115057876037472306157906423, −3.30567502074472137717414886927, −2.53579981867651306039958508650, −2.50276823803824224551266143531, −2.39960404982485733786715156638, −2.15120990843273350105846486230, −1.68278238994317557239708880295, −1.49355668061068021846717217508, −1.19623294082660006707349148348, −0.62352581789061782803589833771, 0.62352581789061782803589833771, 1.19623294082660006707349148348, 1.49355668061068021846717217508, 1.68278238994317557239708880295, 2.15120990843273350105846486230, 2.39960404982485733786715156638, 2.50276823803824224551266143531, 2.53579981867651306039958508650, 3.30567502074472137717414886927, 3.54115057876037472306157906423, 3.62329009957845013216448332342, 3.83289836781046076663451538097, 4.37721507507315872049635166110, 4.38935634780136278507682049183, 4.44310118815521598289723043565, 4.87869756363237281080463613951, 5.21088399746162834604688941721, 5.34461845825756923269238284504, 5.56177397386920940660470103374, 5.67939968546028666580925107230, 6.14045354495691804323772519623, 6.28702418207927132833153110643, 6.66371169786463120612432130959, 6.81067793268610892919957775478, 7.30176268630976462455590035633

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