Properties

Label 8-162e4-1.1-c8e4-0-0
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $1.89693\times 10^{7}$
Root an. cond. $8.12375$
Motivic weight $8$
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  + 128·4-s + 7.06e3·7-s + 8.36e4·13-s − 1.45e5·19-s − 7.26e5·25-s + 9.04e5·28-s + 9.42e5·31-s − 1.20e7·37-s − 7.24e6·43-s + 2.40e7·49-s + 1.07e7·52-s + 1.08e7·61-s − 2.09e6·64-s + 1.22e7·67-s − 1.96e8·73-s − 1.85e7·76-s − 1.67e7·79-s + 5.90e8·91-s − 4.08e7·97-s − 9.30e7·100-s + 5.96e7·103-s − 1.95e8·109-s − 2.15e7·121-s + 1.20e8·124-s + 127-s + 131-s − 1.02e9·133-s + ⋯
L(s)  = 1  + 1/2·4-s + 2.94·7-s + 2.92·13-s − 1.11·19-s − 1.86·25-s + 1.47·28-s + 1.02·31-s − 6.41·37-s − 2.11·43-s + 4.16·49-s + 1.46·52-s + 0.785·61-s − 1/8·64-s + 0.607·67-s − 6.90·73-s − 0.557·76-s − 0.429·79-s + 8.61·91-s − 0.461·97-s − 0.930·100-s + 0.529·103-s − 1.38·109-s − 0.100·121-s + 0.510·124-s − 3.27·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.89693\times 10^{7}\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 4, 4, 4, 4 ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.008772654223\)
\(L(\frac12)\) \(\approx\) \(0.008772654223\)
\(L(5)\) 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$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
3 \( 1 \)
good5$C_2^3$ \( 1 + 29072 p^{2} T^{2} + 601040559 p^{4} T^{4} + 29072 p^{18} T^{6} + p^{32} T^{8} \)
7$C_2^2$ \( ( 1 - 3532 T + 6710223 T^{2} - 3532 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 21566114 T^{2} - 45484632590511165 T^{4} + 21566114 p^{16} T^{6} + p^{32} T^{8} \)
13$C_2^2$ \( ( 1 - 41824 T + 933516255 T^{2} - 41824 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 4967349824 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 36304 T + p^{8} T^{2} )^{4} \)
23$C_2^3$ \( 1 - 14462450590 T^{2} - \)\(59\!\cdots\!61\)\( T^{4} - 14462450590 p^{16} T^{6} + p^{32} T^{8} \)
29$C_2^3$ \( 1 + 928028865104 T^{2} + \)\(61\!\cdots\!95\)\( T^{4} + 928028865104 p^{16} T^{6} + p^{32} T^{8} \)
31$C_2^2$ \( ( 1 - 471196 T - 630865367025 T^{2} - 471196 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 3007402 T + p^{8} T^{2} )^{4} \)
41$C_2^3$ \( 1 + 13027466643584 T^{2} + \)\(10\!\cdots\!15\)\( T^{4} + 13027466643584 p^{16} T^{6} + p^{32} T^{8} \)
43$C_2^2$ \( ( 1 + 3623720 T + 1443146360799 T^{2} + 3623720 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 11446895562722 T^{2} - \)\(43\!\cdots\!37\)\( T^{4} + 11446895562722 p^{16} T^{6} + p^{32} T^{8} \)
53$C_2^2$ \( ( 1 - 18909552109520 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 286434976404290 T^{2} + \)\(60\!\cdots\!59\)\( T^{4} + 286434976404290 p^{16} T^{6} + p^{32} T^{8} \)
61$C_2^2$ \( ( 1 - 5440630 T - 162106858200381 T^{2} - 5440630 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 6121576 T - 368593984832865 T^{2} - 6121576 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 842318136694370 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 49031152 T + p^{8} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 8357756 T - 1447256724551025 T^{2} + 8357756 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 1863467106641954 T^{2} - \)\(16\!\cdots\!65\)\( T^{4} + 1863467106641954 p^{16} T^{6} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 + 3648102662700160 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 20431328 T - 7419994430533377 T^{2} + 20431328 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−7.971371501744652543209132209565, −7.61144252629528010938643465957, −7.25022666923371153021473963175, −7.15741994428207052551993336596, −6.77920982653387373527563812178, −6.46297224512969415332632360089, −6.11155617419926936640205377094, −5.94027331688397822835962678751, −5.36977384135486571567675055666, −5.33108743474238493770738208658, −5.22260264765886095625149082589, −4.55497345522186496811041855296, −4.30723050685683662143424389026, −4.21833726137295965009447905953, −3.74768232879616526365292333773, −3.32016597667790960233876708569, −3.26835024369286242636142376726, −2.73679093895844601324093149332, −1.91500594348254091648637025320, −1.76667096195626605720258105394, −1.69494587604474494071060505663, −1.64184927315158007312631400048, −1.21752319812057277565268710492, −0.65645428640558675201804515367, −0.008486098359518228685904085067, 0.008486098359518228685904085067, 0.65645428640558675201804515367, 1.21752319812057277565268710492, 1.64184927315158007312631400048, 1.69494587604474494071060505663, 1.76667096195626605720258105394, 1.91500594348254091648637025320, 2.73679093895844601324093149332, 3.26835024369286242636142376726, 3.32016597667790960233876708569, 3.74768232879616526365292333773, 4.21833726137295965009447905953, 4.30723050685683662143424389026, 4.55497345522186496811041855296, 5.22260264765886095625149082589, 5.33108743474238493770738208658, 5.36977384135486571567675055666, 5.94027331688397822835962678751, 6.11155617419926936640205377094, 6.46297224512969415332632360089, 6.77920982653387373527563812178, 7.15741994428207052551993336596, 7.25022666923371153021473963175, 7.61144252629528010938643465957, 7.971371501744652543209132209565

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