Properties

Label 8-2160e4-1.1-c1e4-0-20
Degree $8$
Conductor $2.177\times 10^{13}$
Sign $1$
Analytic cond. $88495.9$
Root an. cond. $4.15303$
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  + 16·13-s − 2·25-s + 8·37-s + 28·49-s + 44·61-s + 40·73-s + 64·97-s + 4·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4.43·13-s − 2/5·25-s + 1.31·37-s + 4·49-s + 5.63·61-s + 4.68·73-s + 6.49·97-s + 0.383·109-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 + 8.30·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(88495.9\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.29826845\)
\(L(\frac12)\) \(\approx\) \(10.29826845\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{4} \)
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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

−6.46475564136412098674118331359, −6.11472519359885869200352135259, −6.06250451855410475564519498036, −5.84956113770167196143365289265, −5.81181949498335120766894350145, −5.30512928718617604262627456682, −5.26143362617949643855069495526, −5.09362433305188909596161715492, −4.91019850318182765546988005680, −4.19466329616951735462025244542, −4.11888334293400507511717988645, −4.08616806344695161465742946602, −3.98556232353403183463140996509, −3.55019434004427301762737108254, −3.44357412037090026222253216604, −3.31611044809946375424304163356, −3.05289261089256111662310783569, −2.29055264224896745237293799100, −2.25045616571239268310439312535, −2.21652973268923967098211629542, −1.94329112336789245849960624355, −1.08967268402571049946421798937, −1.05433488004310442440802175617, −0.879090882508719592218855567074, −0.68750312518704842498304116880, 0.68750312518704842498304116880, 0.879090882508719592218855567074, 1.05433488004310442440802175617, 1.08967268402571049946421798937, 1.94329112336789245849960624355, 2.21652973268923967098211629542, 2.25045616571239268310439312535, 2.29055264224896745237293799100, 3.05289261089256111662310783569, 3.31611044809946375424304163356, 3.44357412037090026222253216604, 3.55019434004427301762737108254, 3.98556232353403183463140996509, 4.08616806344695161465742946602, 4.11888334293400507511717988645, 4.19466329616951735462025244542, 4.91019850318182765546988005680, 5.09362433305188909596161715492, 5.26143362617949643855069495526, 5.30512928718617604262627456682, 5.81181949498335120766894350145, 5.84956113770167196143365289265, 6.06250451855410475564519498036, 6.11472519359885869200352135259, 6.46475564136412098674118331359

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