Properties

Label 8-507e4-1.1-c1e4-0-2
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

\(L(1)\) \(\approx\) \(0.2814745716\)
\(L(\frac12)\) \(\approx\) \(0.2814745716\)
\(L(\frac{3}{2})\) 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

−7.78076408619146955936590057979, −7.70566186509270676498625769933, −7.47996980058619059698468913206, −6.81655458109783170326671294122, −6.73367803584789186798194326797, −6.67278585731478039800933820982, −6.48410613122804632727448893041, −6.47988386140100813457565262383, −5.72362514744095755516551829282, −5.71963794326318497165705828476, −5.65585671774579207312841732209, −5.18522340265245109767038366569, −4.91809626670092638930892445213, −4.66501686846975105320051501208, −4.13466662667870497067700669186, −3.82135952072599015126685887052, −3.59663957312564844909724838840, −3.34127778445613368961374692900, −3.27668625423439849235858893490, −2.70460698038876625813352243335, −2.55019411188280531281553922488, −2.32081466924551677437863181310, −1.52052355798266844546797486705, −0.991659761116170447302090637591, −0.20211704338225197482190287919, 0.20211704338225197482190287919, 0.991659761116170447302090637591, 1.52052355798266844546797486705, 2.32081466924551677437863181310, 2.55019411188280531281553922488, 2.70460698038876625813352243335, 3.27668625423439849235858893490, 3.34127778445613368961374692900, 3.59663957312564844909724838840, 3.82135952072599015126685887052, 4.13466662667870497067700669186, 4.66501686846975105320051501208, 4.91809626670092638930892445213, 5.18522340265245109767038366569, 5.65585671774579207312841732209, 5.71963794326318497165705828476, 5.72362514744095755516551829282, 6.47988386140100813457565262383, 6.48410613122804632727448893041, 6.67278585731478039800933820982, 6.73367803584789186798194326797, 6.81655458109783170326671294122, 7.47996980058619059698468913206, 7.70566186509270676498625769933, 7.78076408619146955936590057979

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