Properties

Label 8-2475e4-1.1-c0e4-0-2
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $2.32771$
Root an. cond. $1.11138$
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 + 9-s − 2·11-s + 16-s + 2·31-s + 2·36-s − 4·44-s + 2·49-s − 2·59-s − 2·64-s − 4·71-s − 8·89-s − 2·99-s + 121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 2·4-s + 9-s − 2·11-s + 16-s + 2·31-s + 2·36-s − 4·44-s + 2·49-s − 2·59-s − 2·64-s − 4·71-s − 8·89-s − 2·99-s + 121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.32771\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.861915007\)
\(L(\frac12)\) \(\approx\) \(1.861915007\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_1$ \( ( 1 + T )^{8} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
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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

−6.59554413612106615183297366653, −6.25186879635549421369336562786, −6.23226947785887894819255443348, −5.93228986506209758714463628409, −5.78497627084798025153033917213, −5.62088639152425492222226990586, −5.25116195982091452614199108643, −5.10602467989208173142266921819, −5.06357021274888346396416816148, −4.53069524800404915178777638018, −4.27260811534546924976057160885, −4.22067478512625066326261983860, −4.19964707562303371141548440401, −3.95710386230969074382491729595, −3.16253534847921224917491391369, −3.12548254049514266161692672447, −2.90835925064399848199713132426, −2.71049156770700380527076515308, −2.63632044038703476769391993008, −2.50666096389941352614683487624, −1.81241593553954901483905139104, −1.74284452052943568260306113996, −1.55548732245944899656194330065, −1.27740552360422936359522531804, −0.53115092429659283065575484937, 0.53115092429659283065575484937, 1.27740552360422936359522531804, 1.55548732245944899656194330065, 1.74284452052943568260306113996, 1.81241593553954901483905139104, 2.50666096389941352614683487624, 2.63632044038703476769391993008, 2.71049156770700380527076515308, 2.90835925064399848199713132426, 3.12548254049514266161692672447, 3.16253534847921224917491391369, 3.95710386230969074382491729595, 4.19964707562303371141548440401, 4.22067478512625066326261983860, 4.27260811534546924976057160885, 4.53069524800404915178777638018, 5.06357021274888346396416816148, 5.10602467989208173142266921819, 5.25116195982091452614199108643, 5.62088639152425492222226990586, 5.78497627084798025153033917213, 5.93228986506209758714463628409, 6.23226947785887894819255443348, 6.25186879635549421369336562786, 6.59554413612106615183297366653

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