Properties

Label 8-39e8-1.1-c3e4-0-2
Degree $8$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $6.48606\times 10^{7}$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·4-s + 17·16-s + 96·17-s − 288·23-s − 240·25-s + 384·29-s + 1.28e3·43-s − 592·49-s − 984·53-s + 288·61-s − 153·64-s − 864·68-s + 4.32e3·79-s + 2.59e3·92-s + 2.16e3·100-s + 1.19e3·103-s + 2.49e3·107-s − 3.45e3·116-s − 2.17e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 9/8·4-s + 0.265·16-s + 1.36·17-s − 2.61·23-s − 1.91·25-s + 2.45·29-s + 4.56·43-s − 1.72·49-s − 2.55·53-s + 0.604·61-s − 0.298·64-s − 1.54·68-s + 6.15·79-s + 2.93·92-s + 2.15·100-s + 1.14·103-s + 2.25·107-s − 2.76·116-s − 1.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(6.48606\times 10^{7}\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.611995071\)
\(L(\frac12)\) \(\approx\) \(3.611995071\)
\(L(\frac{5}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 + 9 T^{2} + p^{6} T^{4} + 9 p^{6} T^{6} + p^{12} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 + 48 p T^{2} + 44302 T^{4} + 48 p^{7} T^{6} + p^{12} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 + 592 T^{2} + 310782 T^{4} + 592 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 2172 T^{2} + 3811270 T^{4} + 2172 p^{6} T^{6} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 - 48 T - 1730 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + 13552 T^{2} + 94862766 T^{4} + 13552 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{4} \)
29$D_{4}$ \( ( 1 - 192 T + 45862 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + 18592 T^{2} + 1722523710 T^{4} + 18592 p^{6} T^{6} + p^{12} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 + 92356 T^{2} + 6285341910 T^{4} + 92356 p^{6} T^{6} + p^{12} T^{8} \)
41$C_2^2 \wr C_2$ \( 1 + 142320 T^{2} + 14548520830 T^{4} + 142320 p^{6} T^{6} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 - 644 T + 250566 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + 385164 T^{2} + 58535996374 T^{4} + 385164 p^{6} T^{6} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 + 492 T + 164158 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + 758940 T^{2} + 227837028550 T^{4} + 758940 p^{6} T^{6} + p^{12} T^{8} \)
61$D_{4}$ \( ( 1 - 144 T + 393094 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 611104 T^{2} + 187596510414 T^{4} + 611104 p^{6} T^{6} + p^{12} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 + 514188 T^{2} + 171350572726 T^{4} + 514188 p^{6} T^{6} + p^{12} T^{8} \)
73$C_2^2 \wr C_2$ \( 1 + 659668 T^{2} + 389934034566 T^{4} + 659668 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 2160 T + 2130910 T^{2} - 2160 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 + 823068 T^{2} + 604381635622 T^{4} + 823068 p^{6} T^{6} + p^{12} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + 2258064 T^{2} + 2268670823038 T^{4} + 2258064 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 + 3513652 T^{2} + 4748412207846 T^{4} + 3513652 p^{6} T^{6} + p^{12} T^{8} \)
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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

−6.27761059688710239072835316572, −6.03227056226892833532949985796, −5.99338064002945197627478154690, −5.82243433727049161973506361052, −5.59157550818188078345752440370, −5.05595570024576076584647712644, −4.95161516443529261689380002837, −4.92789887017885145648806710783, −4.75565083510645587656220721721, −4.11207453878641751891100709910, −4.07781631222911387694578150741, −3.97478438547376408660822377134, −3.91381682397942018357553417481, −3.47099208295994480516293997481, −3.18473139534580197253669217107, −2.81321100793184268722843225666, −2.71586862987272771262392252407, −2.37907505748095102302664250422, −1.92855312603084206193136482107, −1.88749426711453453609746327506, −1.48814742020961620584734487152, −1.09659231418290510296333234175, −0.66467533455827391312761878823, −0.54155455502002189805770486356, −0.30225270113315846921080493552, 0.30225270113315846921080493552, 0.54155455502002189805770486356, 0.66467533455827391312761878823, 1.09659231418290510296333234175, 1.48814742020961620584734487152, 1.88749426711453453609746327506, 1.92855312603084206193136482107, 2.37907505748095102302664250422, 2.71586862987272771262392252407, 2.81321100793184268722843225666, 3.18473139534580197253669217107, 3.47099208295994480516293997481, 3.91381682397942018357553417481, 3.97478438547376408660822377134, 4.07781631222911387694578150741, 4.11207453878641751891100709910, 4.75565083510645587656220721721, 4.92789887017885145648806710783, 4.95161516443529261689380002837, 5.05595570024576076584647712644, 5.59157550818188078345752440370, 5.82243433727049161973506361052, 5.99338064002945197627478154690, 6.03227056226892833532949985796, 6.27761059688710239072835316572

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