Properties

Label 8-504e4-1.1-c1e4-0-4
Degree $8$
Conductor $64524128256$
Sign $1$
Analytic cond. $262.319$
Root an. cond. $2.00610$
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  + 2-s + 4-s − 4·7-s + 3·8-s − 4·14-s + 16-s − 8·17-s − 4·23-s + 6·25-s − 4·28-s − 8·31-s − 32-s − 8·34-s + 16·41-s − 4·46-s + 10·49-s + 6·50-s − 12·56-s − 8·62-s + 64-s − 8·68-s + 32·71-s − 24·73-s + 16·82-s + 32·89-s − 4·92-s + 16·97-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 1.06·8-s − 1.06·14-s + 1/4·16-s − 1.94·17-s − 0.834·23-s + 6/5·25-s − 0.755·28-s − 1.43·31-s − 0.176·32-s − 1.37·34-s + 2.49·41-s − 0.589·46-s + 10/7·49-s + 0.848·50-s − 1.60·56-s − 1.01·62-s + 1/8·64-s − 0.970·68-s + 3.79·71-s − 2.80·73-s + 1.76·82-s + 3.39·89-s − 0.417·92-s + 1.62·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(262.319\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.143183454\)
\(L(\frac12)\) \(\approx\) \(2.143183454\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 - 24 T^{2} + 318 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 38 T^{2} + 682 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
19$C_2^2 \wr C_2$ \( 1 - 66 T^{2} + 1794 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2 \wr C_2$ \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2 \wr C_2$ \( 1 - 108 T^{2} + 5382 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 - 152 T^{2} + 9406 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2 \wr C_2$ \( 1 - 132 T^{2} + 8886 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - 178 T^{2} + 14050 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2 \wr C_2$ \( 1 - 230 T^{2} + 20650 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2 \wr C_2$ \( 1 + 112 T^{2} + 8782 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2 \wr C_2$ \( 1 - 210 T^{2} + 23970 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.916875614428652147566611606294, −7.56964502125198251144861050016, −7.18048868992637040371945385627, −7.08379659373969200284921292626, −6.83693683311096910073957549175, −6.81843562019810327053091460628, −6.49992043484608572877308968968, −6.09967830154424433811735931969, −5.87049560584157886858600442569, −5.69862634961928547763427234636, −5.63887132278701090706981106817, −4.93144766077815266352584505483, −4.84505366950805220155504978134, −4.50664624541981976329575105882, −4.38478896978038924125998168348, −4.08247276020000443687536122688, −3.62556719090910256772620378733, −3.45478338536891878295698565254, −3.30546226600309459997193306610, −2.57132012261913177191749122072, −2.52479129934916017308338430212, −2.10898042194840183256146589242, −1.90591324603907588587013517640, −1.10251923814389671908805055877, −0.44934701127787384345848456294, 0.44934701127787384345848456294, 1.10251923814389671908805055877, 1.90591324603907588587013517640, 2.10898042194840183256146589242, 2.52479129934916017308338430212, 2.57132012261913177191749122072, 3.30546226600309459997193306610, 3.45478338536891878295698565254, 3.62556719090910256772620378733, 4.08247276020000443687536122688, 4.38478896978038924125998168348, 4.50664624541981976329575105882, 4.84505366950805220155504978134, 4.93144766077815266352584505483, 5.63887132278701090706981106817, 5.69862634961928547763427234636, 5.87049560584157886858600442569, 6.09967830154424433811735931969, 6.49992043484608572877308968968, 6.81843562019810327053091460628, 6.83693683311096910073957549175, 7.08379659373969200284921292626, 7.18048868992637040371945385627, 7.56964502125198251144861050016, 7.916875614428652147566611606294

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