Properties

Label 8-384e4-1.1-c3e4-0-3
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
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  − 46·9-s − 500·25-s + 1.37e3·49-s + 1.72e3·73-s + 1.38e3·81-s − 7.64e3·97-s + 4.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2.30e4·225-s + ⋯
L(s)  = 1  − 1.70·9-s − 4·25-s + 4·49-s + 2.75·73-s + 1.90·81-s − 7.99·97-s + 3.51·121-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 + 0.000463·167-s + 4·169-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 + 0.000326·211-s + 0.000300·223-s + 6.81·225-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.7753072451\)
\(L(\frac12)\) \(\approx\) \(0.7753072451\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3$C_2^2$ \( 1 + 46 T^{2} + p^{6} T^{4} \)
good5$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2338 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
17$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2}( 1 + 90 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - 2482 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
41$C_2$ \( ( 1 - 522 T + p^{3} T^{2} )^{2}( 1 + 522 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - 74914 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
53$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 304958 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 596626 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 - 430 T + p^{3} T^{2} )^{4} \)
79$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 678926 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 1026 T + p^{3} T^{2} )^{2}( 1 + 1026 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 1910 T + p^{3} T^{2} )^{4} \)
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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

−7.933727857767893288667867694332, −7.50513498166158157610022203528, −7.28865293552701875292825605761, −7.09107231736497162640078807037, −6.77005270257119978712905172653, −6.48448745771862606971275288422, −6.16634286078935731291759537324, −5.77824610563574928417970298036, −5.72773760055005578529838852379, −5.58079139905286779157066356359, −5.36909914044477821358717535262, −5.06102458415549858583442647704, −4.39714878560795207867375465301, −4.32560239626575882758833640778, −3.87032060020497671754576868261, −3.84021682710577073892891907720, −3.51666678436270297953652596006, −2.90019593020829020969599279552, −2.77506563244789333450532936977, −2.35459173960219017974957033223, −2.05384414727084546337938854870, −1.79593205692884855498555751140, −1.12219058602106780854641354648, −0.61095474250302536327077017672, −0.18303542954187130978219015007, 0.18303542954187130978219015007, 0.61095474250302536327077017672, 1.12219058602106780854641354648, 1.79593205692884855498555751140, 2.05384414727084546337938854870, 2.35459173960219017974957033223, 2.77506563244789333450532936977, 2.90019593020829020969599279552, 3.51666678436270297953652596006, 3.84021682710577073892891907720, 3.87032060020497671754576868261, 4.32560239626575882758833640778, 4.39714878560795207867375465301, 5.06102458415549858583442647704, 5.36909914044477821358717535262, 5.58079139905286779157066356359, 5.72773760055005578529838852379, 5.77824610563574928417970298036, 6.16634286078935731291759537324, 6.48448745771862606971275288422, 6.77005270257119978712905172653, 7.09107231736497162640078807037, 7.28865293552701875292825605761, 7.50513498166158157610022203528, 7.933727857767893288667867694332

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