Properties

Label 8-363e4-1.1-c3e4-0-1
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $210421.$
Root an. cond. $4.62792$
Motivic weight $3$
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  − 32·4-s + 54·9-s + 640·16-s + 500·25-s − 1.72e3·36-s − 1.02e4·64-s + 2.18e3·81-s + 5.32e3·97-s − 1.60e4·100-s − 7.28e3·103-s + 127-s + 131-s + 137-s + 139-s + 3.45e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4·4-s + 2·9-s + 10·16-s + 4·25-s − 8·36-s − 20·64-s + 3·81-s + 5.56·97-s − 16·100-s − 6.96·103-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 20·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(210421.\)
Root analytic conductor: \(4.62792\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.5567310503\)
\(L(\frac12)\) \(\approx\) \(0.5567310503\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
5$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
7$C_2^3$ \( 1 - 153502 T^{4} + p^{12} T^{8} \)
13$C_2^3$ \( 1 - 9397582 T^{4} + p^{12} T^{8} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
19$C_2^3$ \( 1 + 17886962 T^{4} + p^{12} T^{8} \)
23$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 35282 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 89206 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
43$C_2^3$ \( 1 - 235885102 T^{4} + p^{12} T^{8} \)
47$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
59$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
61$C_2^3$ \( 1 + 74063873522 T^{4} + p^{12} T^{8} \)
67$C_2^2$ \( ( 1 + 172874 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 104459767778 T^{4} + p^{12} T^{8} \)
79$C_2^3$ \( 1 - 444304748158 T^{4} + p^{12} T^{8} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
89$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )^{4} \)
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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

−7.966513516089296451176106327324, −7.57383814054541680944045456508, −7.43651241022842113863887279740, −7.32211548826783510032045636578, −6.83600830449395616095794306813, −6.44937778627457919169886173626, −6.41355293319915713848900391033, −5.93298115846839608052976937436, −5.53468547051807394727332650086, −5.28284082451617318747288287678, −4.98144706421078158334297850517, −4.89759195854011223711833974442, −4.61801868039802110459195703780, −4.54358563287240891589502108118, −4.20463429007348449522662626190, −3.69173240750259345159717529938, −3.64334611705729364130927708868, −3.53882045588244694535592631610, −2.77816314906968923209518416535, −2.71189489627746504902008844863, −1.71025763493768654281140560058, −1.17374380725163327511973340133, −1.05311777915040532849949489657, −0.872223527049875535865723520176, −0.17607389027743576408005469041, 0.17607389027743576408005469041, 0.872223527049875535865723520176, 1.05311777915040532849949489657, 1.17374380725163327511973340133, 1.71025763493768654281140560058, 2.71189489627746504902008844863, 2.77816314906968923209518416535, 3.53882045588244694535592631610, 3.64334611705729364130927708868, 3.69173240750259345159717529938, 4.20463429007348449522662626190, 4.54358563287240891589502108118, 4.61801868039802110459195703780, 4.89759195854011223711833974442, 4.98144706421078158334297850517, 5.28284082451617318747288287678, 5.53468547051807394727332650086, 5.93298115846839608052976937436, 6.41355293319915713848900391033, 6.44937778627457919169886173626, 6.83600830449395616095794306813, 7.32211548826783510032045636578, 7.43651241022842113863887279740, 7.57383814054541680944045456508, 7.966513516089296451176106327324

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