Properties

Label 8-2520e4-1.1-c0e4-0-0
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  + 2·2-s + 4-s − 2·5-s − 2·8-s − 4·10-s − 4·16-s − 2·19-s − 2·20-s + 2·23-s + 25-s − 2·32-s − 4·38-s + 4·40-s + 4·46-s − 2·47-s − 2·49-s + 2·50-s − 2·53-s + 3·64-s − 2·76-s + 8·80-s + 2·92-s − 4·94-s + 4·95-s − 4·98-s + 100-s − 4·106-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·5-s − 2·8-s − 4·10-s − 4·16-s − 2·19-s − 2·20-s + 2·23-s + 25-s − 2·32-s − 4·38-s + 4·40-s + 4·46-s − 2·47-s − 2·49-s + 2·50-s − 2·53-s + 3·64-s − 2·76-s + 8·80-s + 2·92-s − 4·94-s + 4·95-s − 4·98-s + 100-s − 4·106-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

\(L(\frac{1}{2})\) \(\approx\) \(0.01548486065\)
\(L(\frac12)\) \(\approx\) \(0.01548486065\)
\(L(1)\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3 \( 1 \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.51814815217135803081316912572, −6.13634965895390290746460696705, −6.04110300891148399010946195427, −6.02996015974499335661973024975, −5.50598818165184724551454028923, −5.38543127815873408457543911654, −5.29499869684488890435674153198, −4.84412260915764077913636537256, −4.65011811867723150749378010539, −4.52800755991690738777534013797, −4.52525704264673422429047343810, −4.43107409423867223107507668463, −4.05914706977121436057489958542, −3.70988321191806048983105925157, −3.48608916360494848136469909798, −3.38632115651785794008771527749, −3.30212153460984912665087715354, −2.98476059432350568431861698592, −2.92491642751667109253475955095, −2.25734861373653594131652818678, −2.10220225959405533417816058198, −2.04725755395559543950119135283, −1.17761286650172589734516317596, −1.13118150223226981639620625182, −0.04275398593791337320123136396, 0.04275398593791337320123136396, 1.13118150223226981639620625182, 1.17761286650172589734516317596, 2.04725755395559543950119135283, 2.10220225959405533417816058198, 2.25734861373653594131652818678, 2.92491642751667109253475955095, 2.98476059432350568431861698592, 3.30212153460984912665087715354, 3.38632115651785794008771527749, 3.48608916360494848136469909798, 3.70988321191806048983105925157, 4.05914706977121436057489958542, 4.43107409423867223107507668463, 4.52525704264673422429047343810, 4.52800755991690738777534013797, 4.65011811867723150749378010539, 4.84412260915764077913636537256, 5.29499869684488890435674153198, 5.38543127815873408457543911654, 5.50598818165184724551454028923, 6.02996015974499335661973024975, 6.04110300891148399010946195427, 6.13634965895390290746460696705, 6.51814815217135803081316912572

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