Properties

Label 8-1620e4-1.1-c0e4-0-3
Degree $8$
Conductor $6.887\times 10^{12}$
Sign $1$
Analytic cond. $0.427256$
Root an. cond. $0.899158$
Motivic weight $0$
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  − 2·4-s + 3·16-s + 25-s + 6·31-s + 2·49-s − 2·61-s − 4·64-s − 6·79-s − 2·100-s − 4·109-s − 2·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s + 25-s + 6·31-s + 2·49-s − 2·61-s − 4·64-s − 6·79-s − 2·100-s − 4·109-s − 2·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.427256\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7058714414\)
\(L(\frac12)\) \(\approx\) \(0.7058714414\)
\(L(1)\) 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 \( 1 \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
good7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−7.02336254890689043658756510105, −6.67641740181480838196053470600, −6.43831203277864237896773909676, −6.11578502090099886394843767376, −6.01801089571756233961904196346, −5.83793345669545772773312487639, −5.68526014918180124435450259358, −5.24645850017407491451162262342, −5.07525055456791805950029516436, −4.95648371716187288054778374371, −4.62437775692355609034279256720, −4.38597471077108214470574992390, −4.35306327982578892167889707513, −4.13788599992158762206052105524, −4.01684760411957592133259578399, −3.53447349436593012891692653401, −3.18234470034622892115425425928, −2.99412046526529264398666233906, −2.78164025556725564333853568695, −2.49858949313619016988106553230, −2.49798537604378542099073949450, −1.43622322828939453511757796115, −1.32091325848621284155959167961, −1.21572576887448141047538343544, −0.60702152034050772066015842305, 0.60702152034050772066015842305, 1.21572576887448141047538343544, 1.32091325848621284155959167961, 1.43622322828939453511757796115, 2.49798537604378542099073949450, 2.49858949313619016988106553230, 2.78164025556725564333853568695, 2.99412046526529264398666233906, 3.18234470034622892115425425928, 3.53447349436593012891692653401, 4.01684760411957592133259578399, 4.13788599992158762206052105524, 4.35306327982578892167889707513, 4.38597471077108214470574992390, 4.62437775692355609034279256720, 4.95648371716187288054778374371, 5.07525055456791805950029516436, 5.24645850017407491451162262342, 5.68526014918180124435450259358, 5.83793345669545772773312487639, 6.01801089571756233961904196346, 6.11578502090099886394843767376, 6.43831203277864237896773909676, 6.67641740181480838196053470600, 7.02336254890689043658756510105

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