Properties

Label 8-44e8-1.1-c0e4-0-0
Degree $8$
Conductor $1.405\times 10^{13}$
Sign $1$
Analytic cond. $0.871464$
Root an. cond. $0.982949$
Motivic weight $0$
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  + 2·5-s − 9-s + 25-s − 2·37-s − 2·45-s − 49-s − 2·53-s + 8·89-s + 2·97-s + 2·113-s + ⋯
L(s)  = 1  + 2·5-s − 9-s + 25-s − 2·37-s − 2·45-s − 49-s − 2·53-s + 8·89-s + 2·97-s + 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(0.871464\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 11^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599520173\)
\(L(\frac12)\) \(\approx\) \(1.599520173\)
\(L(1)\) 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 \( 1 \)
11 \( 1 \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_1$ \( ( 1 - T )^{8} \)
97$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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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

−6.71331016882677938413815478260, −6.45762337152535848706026644966, −6.16751525478329651278275421926, −6.15748568949449400737520160146, −5.82207444987546205533785757317, −5.75848180293032728282588841598, −5.70881862668130186777481095734, −5.21161221062349062698801480406, −5.02614575841449739585794271827, −4.85973323335521222352093999177, −4.69609585490604202183230775871, −4.65808734410828753826416517646, −4.15489039082687254128819304982, −3.71572593525804626622560237441, −3.52871041312117170779913686445, −3.38536251320907929983731987403, −3.24126915348585452078129591020, −2.92233788333977228726797664026, −2.57329974492798289371383538778, −2.18490455960584567185170656406, −2.01072964471609107906880497169, −1.82944126276248611970861440434, −1.78203430678774296156021157945, −1.11010788754681871727495403493, −0.65309050940561283484746239137, 0.65309050940561283484746239137, 1.11010788754681871727495403493, 1.78203430678774296156021157945, 1.82944126276248611970861440434, 2.01072964471609107906880497169, 2.18490455960584567185170656406, 2.57329974492798289371383538778, 2.92233788333977228726797664026, 3.24126915348585452078129591020, 3.38536251320907929983731987403, 3.52871041312117170779913686445, 3.71572593525804626622560237441, 4.15489039082687254128819304982, 4.65808734410828753826416517646, 4.69609585490604202183230775871, 4.85973323335521222352093999177, 5.02614575841449739585794271827, 5.21161221062349062698801480406, 5.70881862668130186777481095734, 5.75848180293032728282588841598, 5.82207444987546205533785757317, 6.15748568949449400737520160146, 6.16751525478329651278275421926, 6.45762337152535848706026644966, 6.71331016882677938413815478260

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