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Properties

Label 8-2312e4-1.1-c0e4-0-1
Degree $8$
Conductor $2.857\times 10^{13}$
Sign $1$
Analytic cond. $1.77247$
Root an. cond. $1.07416$
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  − 2·9-s + 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s + 4·97-s − 8·99-s − 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·9-s + 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s + 4·97-s − 8·99-s − 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(1.77247\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.575660539\)
\(L(\frac12)\) \(\approx\) \(1.575660539\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
17 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
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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

−6.50297816248052004954884610195, −6.32648605831513192826945839931, −6.27438554496971190063435593652, −6.15325166178161524119264381249, −5.82191362167887475655326859188, −5.54070834202811906911223800336, −5.53092259209884304349413457502, −5.20897275966266805864975152228, −4.72662067682250487955245300164, −4.59056626578755433530239736212, −4.48876834213203390565649661375, −4.29289097517612842711998691876, −4.11117114495189632421553558691, −3.61010819202117201767602411509, −3.59478408545825812857662888340, −3.42157201302241303239136635156, −3.23508923210504710431662780632, −2.75412850304876361118392196069, −2.57156999374206800551624852836, −2.31740177464408923026737101308, −1.96139932877948714722074248152, −1.79918209705109243525419750463, −1.25969412431646986805119447180, −1.08295442231937820697243018069, −0.67291955124161458460213218934, 0.67291955124161458460213218934, 1.08295442231937820697243018069, 1.25969412431646986805119447180, 1.79918209705109243525419750463, 1.96139932877948714722074248152, 2.31740177464408923026737101308, 2.57156999374206800551624852836, 2.75412850304876361118392196069, 3.23508923210504710431662780632, 3.42157201302241303239136635156, 3.59478408545825812857662888340, 3.61010819202117201767602411509, 4.11117114495189632421553558691, 4.29289097517612842711998691876, 4.48876834213203390565649661375, 4.59056626578755433530239736212, 4.72662067682250487955245300164, 5.20897275966266805864975152228, 5.53092259209884304349413457502, 5.54070834202811906911223800336, 5.82191362167887475655326859188, 6.15325166178161524119264381249, 6.27438554496971190063435593652, 6.32648605831513192826945839931, 6.50297816248052004954884610195

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