Properties

Label 8-507e4-1.1-c1e4-0-11
Degree $8$
Conductor $66074188401$
Sign $1$
Analytic cond. $268.621$
Root an. cond. $2.01206$
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  + 8·7-s − 3·9-s + 4·16-s + 2·19-s − 14·31-s + 22·37-s + 6·43-s + 23·49-s − 24·63-s − 10·67-s − 34·73-s + 28·97-s + 38·109-s + 32·112-s + 127-s + 131-s + 16·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·171-s + 173-s + ⋯
L(s)  = 1  + 3.02·7-s − 9-s + 16-s + 0.458·19-s − 2.51·31-s + 3.61·37-s + 0.914·43-s + 23/7·49-s − 3.02·63-s − 1.22·67-s − 3.97·73-s + 2.84·97-s + 3.63·109-s + 3.02·112-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.458·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(268.621\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.659169431\)
\(L(\frac12)\) \(\approx\) \(3.659169431\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13 \( 1 \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \)
5$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} ) \)
41$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
61$C_2^2$$\times$$C_2^2$ \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} ) \)
71$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 167 T^{2} + p^{2} T^{4} ) \)
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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

−7.84666991913596832879564965631, −7.59545528227057060056025370739, −7.40513329792198021013826116517, −7.37976779402269249295191079346, −7.31928289946748772113047842652, −6.63857292807518547480113441025, −6.11197939753011840303256724559, −6.06601186471511997225804557721, −6.02411790243277064400266106181, −5.54991058153320351694422923174, −5.44782259795638836354684685465, −5.17789256277270035477392217238, −4.77091360016518354446522879715, −4.63997093429983020358734671442, −4.32240127111347398490904268138, −4.26394308657462832395167516331, −3.73642793903640894310048694360, −3.31909925986153169362519755837, −3.12054168560673643210740464439, −2.73470212527227950490945767540, −2.25594546526124639594162286500, −2.02110980973437629670988681889, −1.49099421647136330000285365921, −1.34884139883707453633209073192, −0.66203109531591936146532843532, 0.66203109531591936146532843532, 1.34884139883707453633209073192, 1.49099421647136330000285365921, 2.02110980973437629670988681889, 2.25594546526124639594162286500, 2.73470212527227950490945767540, 3.12054168560673643210740464439, 3.31909925986153169362519755837, 3.73642793903640894310048694360, 4.26394308657462832395167516331, 4.32240127111347398490904268138, 4.63997093429983020358734671442, 4.77091360016518354446522879715, 5.17789256277270035477392217238, 5.44782259795638836354684685465, 5.54991058153320351694422923174, 6.02411790243277064400266106181, 6.06601186471511997225804557721, 6.11197939753011840303256724559, 6.63857292807518547480113441025, 7.31928289946748772113047842652, 7.37976779402269249295191079346, 7.40513329792198021013826116517, 7.59545528227057060056025370739, 7.84666991913596832879564965631

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