Properties

Label 8-2736e4-1.1-c1e4-0-1
Degree $8$
Conductor $5.604\times 10^{13}$
Sign $1$
Analytic cond. $227810.$
Root an. cond. $4.67408$
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  + 4·7-s + 6·13-s − 12·17-s − 16·19-s − 8·25-s − 2·43-s − 12·47-s − 6·49-s + 24·59-s + 2·61-s + 6·67-s + 12·71-s + 14·73-s − 18·79-s + 24·91-s − 36·97-s + 36·101-s + 24·107-s − 48·119-s + 4·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.66·13-s − 2.91·17-s − 3.67·19-s − 8/5·25-s − 0.304·43-s − 1.75·47-s − 6/7·49-s + 3.12·59-s + 0.256·61-s + 0.733·67-s + 1.42·71-s + 1.63·73-s − 2.02·79-s + 2.51·91-s − 3.65·97-s + 3.58·101-s + 2.32·107-s − 4.40·119-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(227810.\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.05846893537\)
\(L(\frac12)\) \(\approx\) \(0.05846893537\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2463 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 2 T - 59 T^{2} - 46 T^{3} + 1948 T^{4} - 46 p T^{5} - 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 12 T + 152 T^{2} + 1248 T^{3} + 10863 T^{4} + 1248 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 24 T + 320 T^{2} - 3312 T^{3} + 28071 T^{4} - 3312 p T^{5} + 320 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T + 77 T^{2} - 390 T^{3} + 540 T^{4} - 390 p T^{5} + 77 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 18 T + 275 T^{2} + 3006 T^{3} + 30180 T^{4} + 3006 p T^{5} + 275 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^3$ \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.28498501606813475465846017659, −6.10298626515778721430123935003, −5.98157773853484316821630077618, −5.81034345443413883564674616634, −5.28931250779234475083679541220, −5.14350651948415531947143695755, −4.98239696342815774093503302745, −4.81860962682089023131791456127, −4.57413233666992765343874673778, −4.39480181769962949658977673957, −4.05461130635036470668516310782, −4.05252640568391062345891931900, −3.83445524858426397345178470961, −3.62940711054893405007587423713, −3.41127217434658786234463635615, −2.95415841897493586841199976714, −2.60255352649336640045602417744, −2.26964039749189624848372912621, −2.12774421401473185936221484510, −2.05283034014670109612566851262, −1.87481873176743822911907625510, −1.46532100779203454900093783724, −1.19262419729026175686604415984, −0.61720684680204542347803852025, −0.04183068726093175586842652065, 0.04183068726093175586842652065, 0.61720684680204542347803852025, 1.19262419729026175686604415984, 1.46532100779203454900093783724, 1.87481873176743822911907625510, 2.05283034014670109612566851262, 2.12774421401473185936221484510, 2.26964039749189624848372912621, 2.60255352649336640045602417744, 2.95415841897493586841199976714, 3.41127217434658786234463635615, 3.62940711054893405007587423713, 3.83445524858426397345178470961, 4.05252640568391062345891931900, 4.05461130635036470668516310782, 4.39480181769962949658977673957, 4.57413233666992765343874673778, 4.81860962682089023131791456127, 4.98239696342815774093503302745, 5.14350651948415531947143695755, 5.28931250779234475083679541220, 5.81034345443413883564674616634, 5.98157773853484316821630077618, 6.10298626515778721430123935003, 6.28498501606813475465846017659

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