Properties

Label 8-2520e4-1.1-c0e4-0-1
Degree $8$
Conductor $4.033\times 10^{13}$
Sign $1$
Analytic cond. $2.50167$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·16-s − 2·49-s − 4·64-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·49-s − 4·64-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2.50167\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4631138334\)
\(L(\frac12)\) \(\approx\) \(0.4631138334\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.41409372856590097726043341641, −6.16345406032983810521421051557, −6.07354163042367560984761999414, −5.87576438843376658651949441175, −5.78306524062205899355110329751, −5.41834087466413072585229649494, −5.13361758867977738076735103171, −4.93841222341856648093892857258, −4.88341680713986146242717727265, −4.80689444032175955156361529305, −4.43369372567518972194928568396, −4.23040721017499197367520859141, −3.98230081103615518005565762796, −3.75818322319115444571216853515, −3.63773937410631060655005934609, −3.24714793227316224933648503366, −3.22677706503195506767150351788, −2.90396793095614900778452243259, −2.57122152479373679057078250940, −2.20431174424225376329667952216, −2.01742807936534045086476165387, −1.50929048096616552089819148766, −1.25481888847999473617822282080, −1.04773547050321697309876268420, −0.36638108559426624345244067375, 0.36638108559426624345244067375, 1.04773547050321697309876268420, 1.25481888847999473617822282080, 1.50929048096616552089819148766, 2.01742807936534045086476165387, 2.20431174424225376329667952216, 2.57122152479373679057078250940, 2.90396793095614900778452243259, 3.22677706503195506767150351788, 3.24714793227316224933648503366, 3.63773937410631060655005934609, 3.75818322319115444571216853515, 3.98230081103615518005565762796, 4.23040721017499197367520859141, 4.43369372567518972194928568396, 4.80689444032175955156361529305, 4.88341680713986146242717727265, 4.93841222341856648093892857258, 5.13361758867977738076735103171, 5.41834087466413072585229649494, 5.78306524062205899355110329751, 5.87576438843376658651949441175, 6.07354163042367560984761999414, 6.16345406032983810521421051557, 6.41409372856590097726043341641

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