Properties

Label 8-384e4-1.1-c1e4-0-2
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $88.3961$
Root an. cond. $1.75107$
Motivic weight $1$
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  − 6·9-s − 4·25-s − 20·49-s + 56·73-s + 27·81-s + 8·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 24·225-s + ⋯
L(s)  = 1  − 2·9-s − 4/5·25-s − 2.85·49-s + 6.55·73-s + 3·81-s + 0.812·97-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 8/5·225-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(88.3961\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.036967883\)
\(L(\frac12)\) \(\approx\) \(1.036967883\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
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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

−8.171050093744916000144528180532, −8.093540629927526297741136445291, −7.87483765279035490706106319678, −7.62000102677691629712604027683, −7.16049473788098287931266862280, −6.97879086546311414503732247931, −6.60591535003832294623708602413, −6.36251806673901999294985814587, −6.31888510608475523802588580068, −5.89490986982345546595031576565, −5.69497222128689072308810440741, −5.42512073171419559766799484655, −5.01786783295086349677149415785, −4.86626467305673063565261434773, −4.80642891446490399157318453175, −4.09027501861879344834821570027, −3.91966465286361320681899352479, −3.41097797971025020596928136546, −3.24064585936274133936720393286, −3.17875123777012981963514068096, −2.36969536973786420608727183049, −2.24922053309418090877101395844, −1.98282690862384287691381763601, −1.14793255303810996878577054749, −0.42577777916060539950582537275, 0.42577777916060539950582537275, 1.14793255303810996878577054749, 1.98282690862384287691381763601, 2.24922053309418090877101395844, 2.36969536973786420608727183049, 3.17875123777012981963514068096, 3.24064585936274133936720393286, 3.41097797971025020596928136546, 3.91966465286361320681899352479, 4.09027501861879344834821570027, 4.80642891446490399157318453175, 4.86626467305673063565261434773, 5.01786783295086349677149415785, 5.42512073171419559766799484655, 5.69497222128689072308810440741, 5.89490986982345546595031576565, 6.31888510608475523802588580068, 6.36251806673901999294985814587, 6.60591535003832294623708602413, 6.97879086546311414503732247931, 7.16049473788098287931266862280, 7.62000102677691629712604027683, 7.87483765279035490706106319678, 8.093540629927526297741136445291, 8.171050093744916000144528180532

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