Properties

Label 8-384e4-1.1-c4e4-0-0
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.48257\times 10^{6}$
Root an. cond. $6.30032$
Motivic weight $4$
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  + 12·3-s − 54·9-s − 536·11-s + 380·25-s − 2.05e3·27-s − 6.43e3·33-s − 6.72e3·49-s + 5.52e3·59-s + 1.95e4·73-s + 4.56e3·75-s − 8.82e3·81-s + 2.36e4·83-s + 2.34e4·97-s + 2.89e4·99-s − 8.50e4·107-s + 1.20e5·121-s + 127-s + 131-s + 137-s + 139-s − 8.06e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.31e4·169-s + ⋯
L(s)  = 1  + 4/3·3-s − 2/3·9-s − 4.42·11-s + 0.607·25-s − 2.81·27-s − 5.90·33-s − 2.80·49-s + 1.58·59-s + 3.67·73-s + 0.810·75-s − 1.34·81-s + 3.43·83-s + 2.49·97-s + 2.95·99-s − 7.43·107-s + 8.26·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 3.73·147-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.26·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.48257\times 10^{6}\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5189364409\)
\(L(\frac12)\) \(\approx\) \(0.5189364409\)
\(L(3)\) 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 \( 1 \)
3$C_2$ \( ( 1 - 2 p T + p^{4} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 38 p T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 3362 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2$ \( ( 1 + 134 T + p^{4} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + 46558 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 110594 T^{2} + p^{8} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 18434 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 144962 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 779522 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 636002 T^{2} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 1156322 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3988034 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 6657602 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3123842 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 13118402 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 1382 T + p^{4} T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 27588002 T^{2} + p^{8} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 40267394 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 47090882 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4894 T + p^{4} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 75709922 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 5914 T + p^{4} T^{2} )^{4} \)
89$C_2$ \( ( 1 - 146 p T + p^{4} T^{2} )^{2}( 1 + 146 p T + p^{4} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5858 T + p^{4} T^{2} )^{4} \)
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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

−7.78332577230208700304679599811, −7.63010264034747106622935490636, −7.10430480391362138649301682272, −6.94236709201376229381263197796, −6.51188032419072245110733329907, −6.20249309461562686878591516958, −6.19745583611674279194586283094, −5.45946633727135401275480781455, −5.45125148743136476628050100445, −5.19985958850412922096076595441, −5.04777850826941420432333828960, −4.90323681184478135873501960048, −4.51948227567807503114102543737, −3.68890912417119097328602323510, −3.64851983149401973492872741171, −3.61305712144318260605117877105, −2.97404287040297724000134835028, −2.66718703119923065581349218555, −2.61632683857268691447442365977, −2.31973833912136690193194768659, −2.22808181711146725598583871078, −1.59186520771417424785865021970, −1.02888819555824534285800994232, −0.39269606082347173082454543518, −0.14272126131083889913452826852, 0.14272126131083889913452826852, 0.39269606082347173082454543518, 1.02888819555824534285800994232, 1.59186520771417424785865021970, 2.22808181711146725598583871078, 2.31973833912136690193194768659, 2.61632683857268691447442365977, 2.66718703119923065581349218555, 2.97404287040297724000134835028, 3.61305712144318260605117877105, 3.64851983149401973492872741171, 3.68890912417119097328602323510, 4.51948227567807503114102543737, 4.90323681184478135873501960048, 5.04777850826941420432333828960, 5.19985958850412922096076595441, 5.45125148743136476628050100445, 5.45946633727135401275480781455, 6.19745583611674279194586283094, 6.20249309461562686878591516958, 6.51188032419072245110733329907, 6.94236709201376229381263197796, 7.10430480391362138649301682272, 7.63010264034747106622935490636, 7.78332577230208700304679599811

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