Properties

Degree $8$
Conductor $1944810000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s + 4·5-s + 6·7-s + 2·9-s + 4·12-s − 8·15-s + 3·16-s + 12·17-s − 8·20-s − 12·21-s + 10·25-s − 6·27-s − 12·28-s + 24·35-s − 4·36-s + 12·37-s + 8·41-s + 16·43-s + 8·45-s − 8·47-s − 6·48-s + 18·49-s − 24·51-s + 16·60-s + 12·63-s − 4·64-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s − 2.06·15-s + 3/4·16-s + 2.91·17-s − 1.78·20-s − 2.61·21-s + 2·25-s − 1.15·27-s − 2.26·28-s + 4.05·35-s − 2/3·36-s + 1.97·37-s + 1.24·41-s + 2.43·43-s + 1.19·45-s − 1.16·47-s − 0.866·48-s + 18/7·49-s − 3.36·51-s + 2.06·60-s + 1.51·63-s − 1/2·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.85122\)
\(L(\frac12)\) \(\approx\) \(1.85122\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 2 T - 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \)
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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

−9.274618619610422215752940428211, −8.938106051143470080906408762661, −8.494393243429295623907305615873, −8.111410762312666473310373948238, −7.919321663811144579775381784087, −7.88102971389047702579319886662, −7.52206610716490829259471236966, −7.21880171673753600146579085201, −6.98616253069062895879214867105, −6.22383389065855848885223247386, −6.21026672455125969005904546329, −5.78020523149887817724904724686, −5.56250779248751694961822887916, −5.49291523962495038756578242293, −5.39378971821176985731350793601, −4.85894466694509251136112359601, −4.37719302623411572614881697721, −4.25110150697653622959183206661, −4.24058733389538653244322784739, −3.33016707163840279440937490097, −2.72901789556295232313364175973, −2.70748439083832360578983353683, −1.70144517042571448310252289478, −1.29525487535678039644420450590, −1.21527653637586685339144657534, 1.21527653637586685339144657534, 1.29525487535678039644420450590, 1.70144517042571448310252289478, 2.70748439083832360578983353683, 2.72901789556295232313364175973, 3.33016707163840279440937490097, 4.24058733389538653244322784739, 4.25110150697653622959183206661, 4.37719302623411572614881697721, 4.85894466694509251136112359601, 5.39378971821176985731350793601, 5.49291523962495038756578242293, 5.56250779248751694961822887916, 5.78020523149887817724904724686, 6.21026672455125969005904546329, 6.22383389065855848885223247386, 6.98616253069062895879214867105, 7.21880171673753600146579085201, 7.52206610716490829259471236966, 7.88102971389047702579319886662, 7.919321663811144579775381784087, 8.111410762312666473310373948238, 8.494393243429295623907305615873, 8.938106051143470080906408762661, 9.274618619610422215752940428211

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