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Properties

Label 8-2624e4-1.1-c0e4-0-2
Degree $8$
Conductor $4.741\times 10^{13}$
Sign $1$
Analytic cond. $2.94092$
Root an. cond. $1.14435$
Motivic weight $0$
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  − 2·9-s − 4·11-s + 4·17-s − 4·19-s + 2·81-s − 4·89-s + 8·99-s + 8·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 8·171-s + 173-s + 179-s + 181-s − 16·187-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·9-s − 4·11-s + 4·17-s − 4·19-s + 2·81-s − 4·89-s + 8·99-s + 8·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 8·171-s + 173-s + 179-s + 181-s − 16·187-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(2.94092\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 41^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4994961282\)
\(L(\frac12)\) \(\approx\) \(0.4994961282\)
\(L(1)\) 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 \( 1 \)
41$C_2^2$ \( 1 + T^{4} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
19$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_4\times C_2$ \( 1 + T^{8} \)
53$C_4\times C_2$ \( 1 + T^{8} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
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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

−6.33963166228637553825639327407, −6.02211789212406739084183808350, −5.95853196698206061210858238053, −5.86171065458443389285808905143, −5.67422828821503697702426719231, −5.61729305777396707331739789346, −5.28623347039256019645673451123, −5.12445320423446849715699457575, −4.89912000462713128644739225252, −4.63216502594911689081916207327, −4.51088297816719601155979824284, −4.35386066678508234214745929378, −3.76382665289968595180903559625, −3.73285571600481136375248208705, −3.31436410538167939806738588133, −3.08685622813472834822695269126, −3.03084794074718442300328371036, −2.92725474688667468610358531362, −2.50028233099838136800029981512, −2.20803697911532689629692635379, −2.13545714669050891870357129307, −1.91734063337439813119574432239, −1.39219138530293750506319819727, −0.70364968470200528764048014780, −0.41296156916163713996300304616, 0.41296156916163713996300304616, 0.70364968470200528764048014780, 1.39219138530293750506319819727, 1.91734063337439813119574432239, 2.13545714669050891870357129307, 2.20803697911532689629692635379, 2.50028233099838136800029981512, 2.92725474688667468610358531362, 3.03084794074718442300328371036, 3.08685622813472834822695269126, 3.31436410538167939806738588133, 3.73285571600481136375248208705, 3.76382665289968595180903559625, 4.35386066678508234214745929378, 4.51088297816719601155979824284, 4.63216502594911689081916207327, 4.89912000462713128644739225252, 5.12445320423446849715699457575, 5.28623347039256019645673451123, 5.61729305777396707331739789346, 5.67422828821503697702426719231, 5.86171065458443389285808905143, 5.95853196698206061210858238053, 6.02211789212406739084183808350, 6.33963166228637553825639327407

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