Properties

Label 8-1400e4-1.1-c0e4-0-3
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
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 + 2·11-s − 2·19-s + 2·36-s − 4·41-s + 2·44-s − 2·49-s + 4·59-s − 64-s − 2·76-s + 81-s + 4·89-s + 4·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 4·171-s + ⋯
L(s)  = 1  + 4-s + 2·9-s + 2·11-s − 2·19-s + 2·36-s − 4·41-s + 2·44-s − 2·49-s + 4·59-s − 64-s − 2·76-s + 81-s + 4·89-s + 4·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 4·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.836422551\)
\(L(\frac12)\) \(\approx\) \(1.836422551\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_2$ \( ( 1 + T + T^{2} )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
59$C_2$ \( ( 1 - T + T^{2} )^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{4} \)
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

−6.80149828697297067313718488760, −6.69366977624220667306758680083, −6.65530479268562245698338828513, −6.62183338378830887072184456514, −6.44410319688415009718417845928, −6.04765220415516302250772712997, −5.70991421540247825909762392288, −5.69772431540682343415447595872, −5.21939050070547900073886043687, −4.79624190615770395210652735169, −4.75170224060472153993523193261, −4.69796863583890074529305885822, −4.46019147862406910841822449111, −3.89944864340438923751048250794, −3.74847151544128625713042617669, −3.65014182070340203261061302255, −3.60539332497162277590541150234, −3.22092628046111606104817532526, −2.56916313280053855079446431611, −2.41174021469771824549129128864, −2.22789076054254731537704122915, −1.64375435082450942846861992013, −1.58105235139067307344594910014, −1.56204309910663500106967594800, −0.859251812738902875620641057435, 0.859251812738902875620641057435, 1.56204309910663500106967594800, 1.58105235139067307344594910014, 1.64375435082450942846861992013, 2.22789076054254731537704122915, 2.41174021469771824549129128864, 2.56916313280053855079446431611, 3.22092628046111606104817532526, 3.60539332497162277590541150234, 3.65014182070340203261061302255, 3.74847151544128625713042617669, 3.89944864340438923751048250794, 4.46019147862406910841822449111, 4.69796863583890074529305885822, 4.75170224060472153993523193261, 4.79624190615770395210652735169, 5.21939050070547900073886043687, 5.69772431540682343415447595872, 5.70991421540247825909762392288, 6.04765220415516302250772712997, 6.44410319688415009718417845928, 6.62183338378830887072184456514, 6.65530479268562245698338828513, 6.69366977624220667306758680083, 6.80149828697297067313718488760

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