Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{12} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 3·5-s − 2·7-s + 2·8-s − 6·10-s + 3·11-s − 4·13-s + 4·14-s − 4·16-s + 6·17-s + 20·19-s + 3·20-s − 6·22-s − 9·23-s + 4·25-s + 8·26-s − 2·28-s − 6·29-s − 4·31-s + 2·32-s − 12·34-s − 6·35-s + 8·37-s − 40·38-s + 6·40-s + 15·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s + 0.707·8-s − 1.89·10-s + 0.904·11-s − 1.10·13-s + 1.06·14-s − 16-s + 1.45·17-s + 4.58·19-s + 0.670·20-s − 1.27·22-s − 1.87·23-s + 4/5·25-s + 1.56·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.353·32-s − 2.05·34-s − 1.01·35-s + 1.31·37-s − 6.48·38-s + 0.948·40-s + 2.34·41-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{378} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.43955$
$L(\frac12)$  $\approx$  $1.43955$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$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$ \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
17$D_{4}$ \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 9 T + p T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} ) \)
29$D_4\times C_2$ \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \)
71$D_{4}$ \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.342269316485315092441089560446, −7.84261529258016309172135468118, −7.62179457492301691383139650658, −7.57753157859405492013002111620, −7.37209212560432781948332279121, −7.22057338555977952145283121708, −7.00372585369883507184161938496, −6.43584126441933528374313608425, −6.06518795392006075871071646948, −6.01462663131127110654953456431, −5.68280733791041058273165830676, −5.51855206258890072817998252812, −5.26204126986445747167639892258, −5.07219607525224819657031538209, −4.51131880856165925700181116171, −4.12592636405168422921552723852, −4.01935908558319669196968515302, −3.33027456357761475730101904344, −3.19346068723694222992876887276, −3.08583661968968612764015978118, −2.24747080160459599453703670913, −2.16959692094955805268421014912, −1.57879843614538794365131137200, −0.925166825961432622441268481514, −0.866087841543724400601761598973, 0.866087841543724400601761598973, 0.925166825961432622441268481514, 1.57879843614538794365131137200, 2.16959692094955805268421014912, 2.24747080160459599453703670913, 3.08583661968968612764015978118, 3.19346068723694222992876887276, 3.33027456357761475730101904344, 4.01935908558319669196968515302, 4.12592636405168422921552723852, 4.51131880856165925700181116171, 5.07219607525224819657031538209, 5.26204126986445747167639892258, 5.51855206258890072817998252812, 5.68280733791041058273165830676, 6.01462663131127110654953456431, 6.06518795392006075871071646948, 6.43584126441933528374313608425, 7.00372585369883507184161938496, 7.22057338555977952145283121708, 7.37209212560432781948332279121, 7.57753157859405492013002111620, 7.62179457492301691383139650658, 7.84261529258016309172135468118, 8.342269316485315092441089560446

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