Properties

Label 8-1134e4-1.1-c1e4-0-16
Degree $8$
Conductor $1.654\times 10^{12}$
Sign $1$
Analytic cond. $6722.96$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 4·5-s + 2·7-s − 2·8-s − 8·10-s − 2·11-s + 6·13-s + 4·14-s − 4·16-s + 28·17-s − 4·19-s − 4·20-s − 4·22-s − 2·23-s + 11·25-s + 12·26-s + 2·28-s − 10·29-s − 6·31-s − 2·32-s + 56·34-s − 8·35-s + 8·37-s − 8·38-s + 8·40-s − 12·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s − 0.707·8-s − 2.52·10-s − 0.603·11-s + 1.66·13-s + 1.06·14-s − 16-s + 6.79·17-s − 0.917·19-s − 0.894·20-s − 0.852·22-s − 0.417·23-s + 11/5·25-s + 2.35·26-s + 0.377·28-s − 1.85·29-s − 1.07·31-s − 0.353·32-s + 9.60·34-s − 1.35·35-s + 1.31·37-s − 1.29·38-s + 1.26·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(6722.96\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.419722957\)
\(L(\frac12)\) \(\approx\) \(6.419722957\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3 \( 1 \)
7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good5$C_2$$\times$$C_2^2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 + 2 T + 8 T^{2} - 52 T^{3} - 149 T^{4} - 52 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
19$D_{4}$ \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
31$D_4\times C_2$ \( 1 + 6 T - 8 T^{2} - 108 T^{3} - 141 T^{4} - 108 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 12 T + 38 T^{2} + 288 T^{3} + 3651 T^{4} + 288 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 4 T - 62 T^{2} - 32 T^{3} + 3547 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 6 T - 64 T^{2} + 36 T^{3} + 6099 T^{4} + 36 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 2 T - 88 T^{2} + 52 T^{3} + 4747 T^{4} + 52 p T^{5} - 88 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 10 T - 56 T^{2} - 220 T^{3} + 13147 T^{4} - 220 p T^{5} - 56 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 20 T + 230 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 6 T + 16 T^{2} - 828 T^{3} - 8685 T^{4} - 828 p T^{5} + 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 2 T + 80 T^{2} - 484 T^{3} - 1445 T^{4} - 484 p T^{5} + 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 8 T - 98 T^{2} - 256 T^{3} + 10627 T^{4} - 256 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.95130547672491767199069962486, −6.86051906035809528771133714065, −6.36666649567413165574410089237, −6.22928616918537220063514105044, −6.20887189513268615434856470524, −5.61846254227166498520150245261, −5.61497502576386169949745503667, −5.28962954857802014753506886918, −5.20618077169820451972536281678, −5.03054689484103839872624904583, −4.98651056292367953502996650695, −4.48972973627724754441536214373, −3.97980651813358106223305815823, −3.95207843706092304167718450726, −3.68132654074708492019258877638, −3.43272112536154252208592572490, −3.38393939780238345551199140049, −3.29655871812561688758453350403, −3.12639200828716152528873062148, −2.36518426852689576464114825423, −2.10469111124698932635189815323, −1.54555749821710919977399146638, −1.31721110419084448794829601550, −0.872218173641868353925790347405, −0.54149728045920001634620874394, 0.54149728045920001634620874394, 0.872218173641868353925790347405, 1.31721110419084448794829601550, 1.54555749821710919977399146638, 2.10469111124698932635189815323, 2.36518426852689576464114825423, 3.12639200828716152528873062148, 3.29655871812561688758453350403, 3.38393939780238345551199140049, 3.43272112536154252208592572490, 3.68132654074708492019258877638, 3.95207843706092304167718450726, 3.97980651813358106223305815823, 4.48972973627724754441536214373, 4.98651056292367953502996650695, 5.03054689484103839872624904583, 5.20618077169820451972536281678, 5.28962954857802014753506886918, 5.61497502576386169949745503667, 5.61846254227166498520150245261, 6.20887189513268615434856470524, 6.22928616918537220063514105044, 6.36666649567413165574410089237, 6.86051906035809528771133714065, 6.95130547672491767199069962486

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