Properties

Label 8-21e8-1.1-c0e4-0-0
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $0.00234629$
Root an. cond. $0.469135$
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  + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.00234629\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5084325896\)
\(L(\frac12)\) \(\approx\) \(0.5084325896\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2^3$ \( 1 - T^{4} + T^{8} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−8.385762160058102779869942834926, −7.889531821328608948275817454481, −7.84273445823638068332700752822, −7.59703851389044336707338211636, −7.51647339343988795042853942915, −6.98762824746807386839413737328, −6.73258232742497397747311064116, −6.65004984613259944318897430329, −6.36194329576796973093361987275, −5.99673600071139469592678931300, −5.63783728162440250647776145816, −5.58395740293661999262320915742, −5.41790831006616825823780459578, −4.91076093647297368638248374155, −4.83744785579779234334522478818, −4.24701759009354368989110402291, −4.03905757834259701265667766827, −3.86931961490569081585999189777, −3.53857529263476151221789652631, −3.25394871834383751516281348352, −2.71155808384978503277722180903, −2.47891726023275405362271729690, −2.14017190261486800664524466824, −1.51270748961077959396563896666, −1.22447069934551854485032977313, 1.22447069934551854485032977313, 1.51270748961077959396563896666, 2.14017190261486800664524466824, 2.47891726023275405362271729690, 2.71155808384978503277722180903, 3.25394871834383751516281348352, 3.53857529263476151221789652631, 3.86931961490569081585999189777, 4.03905757834259701265667766827, 4.24701759009354368989110402291, 4.83744785579779234334522478818, 4.91076093647297368638248374155, 5.41790831006616825823780459578, 5.58395740293661999262320915742, 5.63783728162440250647776145816, 5.99673600071139469592678931300, 6.36194329576796973093361987275, 6.65004984613259944318897430329, 6.73258232742497397747311064116, 6.98762824746807386839413737328, 7.51647339343988795042853942915, 7.59703851389044336707338211636, 7.84273445823638068332700752822, 7.889531821328608948275817454481, 8.385762160058102779869942834926

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