Properties

Label 8-11e8-1.1-c3e4-0-0
Degree $8$
Conductor $214358881$
Sign $1$
Analytic cond. $2597.80$
Root an. cond. $2.67193$
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  − 8·3-s + 8·4-s − 18·5-s + 27·9-s − 64·12-s + 144·15-s − 144·20-s − 432·23-s + 125·25-s − 340·31-s + 216·36-s + 434·37-s − 486·45-s + 36·47-s + 343·49-s + 738·53-s + 720·59-s + 1.15e3·60-s − 1.66e3·67-s + 3.45e3·69-s − 612·71-s − 1.00e3·75-s + 6.69e3·89-s − 3.45e3·92-s + 2.72e3·93-s + 34·97-s + 1.00e3·100-s + ⋯
L(s)  = 1  − 1.53·3-s + 4-s − 1.60·5-s + 9-s − 1.53·12-s + 2.47·15-s − 1.60·20-s − 3.91·23-s + 25-s − 1.96·31-s + 36-s + 1.92·37-s − 1.60·45-s + 0.111·47-s + 49-s + 1.91·53-s + 1.58·59-s + 2.47·60-s − 3.03·67-s + 6.02·69-s − 1.02·71-s − 1.53·75-s + 7.97·89-s − 3.91·92-s + 3.03·93-s + 0.0355·97-s + 100-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(11^{8}\)
Sign: $1$
Analytic conductor: \(2597.80\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.5995704265\)
\(L(\frac12)\) \(\approx\) \(0.5995704265\)
\(L(\frac{5}{2})\) 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)$
bad11 \( 1 \)
good2$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
3$C_4\times C_2$ \( 1 + 8 T + 37 T^{2} + 80 T^{3} - 359 T^{4} + 80 p^{3} T^{5} + 37 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
5$C_4\times C_2$ \( 1 + 18 T + 199 T^{2} + 1332 T^{3} - 899 T^{4} + 1332 p^{3} T^{5} + 199 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \)
7$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
13$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
17$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
19$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
23$C_2$ \( ( 1 + 108 T + p^{3} T^{2} )^{4} \)
29$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
31$C_4\times C_2$ \( 1 + 340 T + 85809 T^{2} + 19046120 T^{3} + 3919344881 T^{4} + 19046120 p^{3} T^{5} + 85809 p^{6} T^{6} + 340 p^{9} T^{7} + p^{12} T^{8} \)
37$C_4\times C_2$ \( 1 - 434 T + 137703 T^{2} - 37779700 T^{3} + 9421319741 T^{4} - 37779700 p^{3} T^{5} + 137703 p^{6} T^{6} - 434 p^{9} T^{7} + p^{12} T^{8} \)
41$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
43$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
47$C_4\times C_2$ \( 1 - 36 T - 102527 T^{2} + 7428600 T^{3} + 10377231121 T^{4} + 7428600 p^{3} T^{5} - 102527 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \)
53$C_4\times C_2$ \( 1 - 738 T + 395767 T^{2} - 182204820 T^{3} + 75546553501 T^{4} - 182204820 p^{3} T^{5} + 395767 p^{6} T^{6} - 738 p^{9} T^{7} + p^{12} T^{8} \)
59$C_4\times C_2$ \( 1 - 720 T + 313021 T^{2} - 77502240 T^{3} - 8486327159 T^{4} - 77502240 p^{3} T^{5} + 313021 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \)
61$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
67$C_2$ \( ( 1 + 416 T + p^{3} T^{2} )^{4} \)
71$C_4\times C_2$ \( 1 + 612 T + 16633 T^{2} - 208862136 T^{3} - 133776760895 T^{4} - 208862136 p^{3} T^{5} + 16633 p^{6} T^{6} + 612 p^{9} T^{7} + p^{12} T^{8} \)
73$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
79$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
83$C_4\times C_2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} \)
89$C_2$ \( ( 1 - 1674 T + p^{3} T^{2} )^{4} \)
97$C_4\times C_2$ \( 1 - 34 T - 911517 T^{2} + 62022460 T^{3} + 829808191301 T^{4} + 62022460 p^{3} T^{5} - 911517 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} \)
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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

−9.472763262618668507949373388747, −9.002458197999781371364427809406, −8.962539283011321833536927259661, −8.533591763917543177240355199418, −7.985445085622730871184504535243, −7.74778074866152736214190724862, −7.74610184048241485852764267269, −7.48632012191810004029955355413, −7.07981167763666819667030106055, −6.63569402112610197700212531436, −6.49318848123488723063215350588, −5.91050881802025028260150712366, −5.90503619536109020927685983675, −5.77514625800853500204102325643, −5.00154385944659121018435388445, −4.92532901675372698611808546685, −4.13539684171945595860860964164, −4.06522403580169364017072692867, −3.87143943570272122254363997370, −3.35508336762068117440483655075, −2.59746914012761409600697914585, −2.15067441588553204296212581698, −1.80673807592092051780541294447, −0.73736457287548767861729506349, −0.31473449184664528444840498007, 0.31473449184664528444840498007, 0.73736457287548767861729506349, 1.80673807592092051780541294447, 2.15067441588553204296212581698, 2.59746914012761409600697914585, 3.35508336762068117440483655075, 3.87143943570272122254363997370, 4.06522403580169364017072692867, 4.13539684171945595860860964164, 4.92532901675372698611808546685, 5.00154385944659121018435388445, 5.77514625800853500204102325643, 5.90503619536109020927685983675, 5.91050881802025028260150712366, 6.49318848123488723063215350588, 6.63569402112610197700212531436, 7.07981167763666819667030106055, 7.48632012191810004029955355413, 7.74610184048241485852764267269, 7.74778074866152736214190724862, 7.985445085622730871184504535243, 8.533591763917543177240355199418, 8.962539283011321833536927259661, 9.002458197999781371364427809406, 9.472763262618668507949373388747

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