Properties

Label 8-210e4-1.1-c1e4-0-0
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
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  + 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: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(1.497125679\)
\(L(\frac12)\) \(\approx\) \(1.497125679\)
\(L(\frac{3}{2})\) 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

−8.944183526097975663103759818318, −8.688746073863802366844478885316, −8.583482131391583359230775298980, −8.373093804957970469888038664717, −7.985756064625003225791222056715, −7.74747172114191661064883955424, −7.55090833525974995408141806590, −7.49038765926742903438116670008, −7.08277001978161789814923756476, −6.82706276134446434623250687926, −6.13616947239813470342391372262, −6.11185740583984162563712274414, −5.79065936957182111881195940933, −4.96317959414577807808508999949, −4.83596773642701562228480550864, −4.68196763603880172432035843732, −4.40572789200858677231059812906, −4.20354530253592862062616267166, −4.01044565414653378996205278864, −3.33634467327170917169484353562, −3.13685333059573182797924868496, −2.36425259544367887198175546683, −2.33004092411596175300282581386, −1.57222858037038003848599506127, −0.74380502191312902225374957915, 0.74380502191312902225374957915, 1.57222858037038003848599506127, 2.33004092411596175300282581386, 2.36425259544367887198175546683, 3.13685333059573182797924868496, 3.33634467327170917169484353562, 4.01044565414653378996205278864, 4.20354530253592862062616267166, 4.40572789200858677231059812906, 4.68196763603880172432035843732, 4.83596773642701562228480550864, 4.96317959414577807808508999949, 5.79065936957182111881195940933, 6.11185740583984162563712274414, 6.13616947239813470342391372262, 6.82706276134446434623250687926, 7.08277001978161789814923756476, 7.49038765926742903438116670008, 7.55090833525974995408141806590, 7.74747172114191661064883955424, 7.985756064625003225791222056715, 8.373093804957970469888038664717, 8.583482131391583359230775298980, 8.688746073863802366844478885316, 8.944183526097975663103759818318

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