Properties

Label 8-1520e4-1.1-c1e4-0-5
Degree $8$
Conductor $5.338\times 10^{12}$
Sign $1$
Analytic cond. $21701.1$
Root an. cond. $3.48385$
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  − 4·5-s + 4·19-s + 2·25-s − 8·29-s + 24·31-s + 32·41-s − 12·49-s − 24·59-s + 24·71-s − 8·79-s + 2·81-s − 8·89-s − 16·95-s + 16·101-s − 44·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s − 96·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.917·19-s + 2/5·25-s − 1.48·29-s + 4.31·31-s + 4.99·41-s − 1.71·49-s − 3.12·59-s + 2.84·71-s − 0.900·79-s + 2/9·81-s − 0.847·89-s − 1.64·95-s + 1.59·101-s − 4·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 7.71·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(21701.1\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.492601044\)
\(L(\frac12)\) \(\approx\) \(2.492601044\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_1$ \( ( 1 - T )^{4} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
31$D_{4}$ \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 40 T^{2} + 2158 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
41$C_4$ \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T^{2} - 602 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 40 T^{2} + 4398 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 256 T^{2} + 25342 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 244 T^{2} + 25222 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 100 T^{2} + 4758 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 40 T^{2} - 5282 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.75748918684056387131139544478, −6.42881252601734286860390947471, −6.38917974441952988610920276054, −6.11796060274431820961882639778, −5.85924386847324735692133598094, −5.78904643638252697454710700331, −5.58091973879745802656206820661, −4.98754678882548136076249449768, −4.83756397347429447969902814539, −4.71061199632861557512404499118, −4.70923249489100859106265713991, −4.08466713788181593607442808809, −4.04581059692740183685151766327, −3.92019161124086504698463518031, −3.77424793644705896764644534687, −3.17061122152241894742194018719, −3.10673657472407400143653004563, −2.80689279243106968751623605986, −2.57464255619718665494147186818, −2.41357424528843728019050976638, −1.82542113696614157635501596830, −1.47694462615177501787957281178, −1.12207663111506141448434994609, −0.55281636755501439907646968997, −0.54003496349931498685870966663, 0.54003496349931498685870966663, 0.55281636755501439907646968997, 1.12207663111506141448434994609, 1.47694462615177501787957281178, 1.82542113696614157635501596830, 2.41357424528843728019050976638, 2.57464255619718665494147186818, 2.80689279243106968751623605986, 3.10673657472407400143653004563, 3.17061122152241894742194018719, 3.77424793644705896764644534687, 3.92019161124086504698463518031, 4.04581059692740183685151766327, 4.08466713788181593607442808809, 4.70923249489100859106265713991, 4.71061199632861557512404499118, 4.83756397347429447969902814539, 4.98754678882548136076249449768, 5.58091973879745802656206820661, 5.78904643638252697454710700331, 5.85924386847324735692133598094, 6.11796060274431820961882639778, 6.38917974441952988610920276054, 6.42881252601734286860390947471, 6.75748918684056387131139544478

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