Properties

Label 8-26e8-1.1-c0e4-0-0
Degree $8$
Conductor $208827064576$
Sign $1$
Analytic cond. $0.0129543$
Root an. cond. $0.580833$
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  + 4-s − 2·9-s − 2·17-s + 2·25-s + 2·29-s − 2·36-s + 2·49-s − 4·53-s + 2·61-s − 64-s − 2·68-s + 81-s + 2·100-s − 2·101-s + 2·113-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4-s − 2·9-s − 2·17-s + 2·25-s + 2·29-s − 2·36-s + 2·49-s − 4·53-s + 2·61-s − 64-s − 2·68-s + 81-s + 2·100-s − 2·101-s + 2·113-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(0.0129543\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{676} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 13^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6819051624\)
\(L(\frac12)\) \(\approx\) \(0.6819051624\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
13 \( 1 \)
good3$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
5$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$ \( ( 1 + T + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−8.033220507636001569377020174735, −7.50173042721762220985295565719, −7.11380780006616797039149253379, −7.01559081977551854781741787617, −6.76022627790139374268002653921, −6.67732483656634020697300820469, −6.52338077753115756403963101395, −6.12952934496471266055220023215, −5.88734466128779035642889046141, −5.71356026498068986380715965128, −5.60607804825502750833664267745, −4.95434884108137021917090736182, −4.83363210795382071037005506306, −4.78696513787114465596855014276, −4.42765494169131592724342385664, −4.16069121188017799988245779962, −3.69714944855823528783316673315, −3.32305686548256830028198989712, −3.08695566288795130996984523323, −2.80508990870431593667542644801, −2.54147346091484273542543942998, −2.45626233076909153801889847746, −1.95662763632202046635197634862, −1.53980948264443174890148283943, −0.852200475815374476339511689986, 0.852200475815374476339511689986, 1.53980948264443174890148283943, 1.95662763632202046635197634862, 2.45626233076909153801889847746, 2.54147346091484273542543942998, 2.80508990870431593667542644801, 3.08695566288795130996984523323, 3.32305686548256830028198989712, 3.69714944855823528783316673315, 4.16069121188017799988245779962, 4.42765494169131592724342385664, 4.78696513787114465596855014276, 4.83363210795382071037005506306, 4.95434884108137021917090736182, 5.60607804825502750833664267745, 5.71356026498068986380715965128, 5.88734466128779035642889046141, 6.12952934496471266055220023215, 6.52338077753115756403963101395, 6.67732483656634020697300820469, 6.76022627790139374268002653921, 7.01559081977551854781741787617, 7.11380780006616797039149253379, 7.50173042721762220985295565719, 8.033220507636001569377020174735

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