Properties

Label 8-2028e4-1.1-c1e4-0-6
Degree $8$
Conductor $1.691\times 10^{13}$
Sign $1$
Analytic cond. $68767.0$
Root an. cond. $4.02413$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s + 4·17-s − 12·25-s − 2·27-s + 12·29-s + 8·43-s − 10·49-s + 8·51-s − 40·53-s + 28·61-s − 24·75-s − 64·79-s − 4·81-s + 24·87-s + 20·101-s + 32·103-s − 24·107-s − 12·113-s − 6·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 20·147-s + 149-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 0.970·17-s − 2.39·25-s − 0.384·27-s + 2.22·29-s + 1.21·43-s − 1.42·49-s + 1.12·51-s − 5.49·53-s + 3.58·61-s − 2.77·75-s − 7.20·79-s − 4/9·81-s + 2.57·87-s + 1.99·101-s + 3.15·103-s − 2.32·107-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(68767.0\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2028} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.339575804\)
\(L(\frac12)\) \(\approx\) \(5.339575804\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
43$C_2^2$ \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^3$ \( 1 - 114 T^{2} + 7955 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2^3$ \( 1 + 162 T^{2} + 18323 T^{4} + 162 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.41828526041226557928924594561, −6.38827513399849217724341860481, −6.20094730782170497679266898390, −5.81123908823528425677135855811, −5.56159385640590190717119270674, −5.55084746919041315040853094391, −5.40827160720220275282425380912, −4.75365138107859220109840666807, −4.75188689505607587174472311312, −4.66340665222888927928543807467, −4.27930073702111663760417087140, −4.02125708518221809874551826142, −3.99351048552983604914924096749, −3.51043025878399467315072527523, −3.28044049645570951462659754632, −3.22844972834352146454980873119, −2.79454867483433052481827070279, −2.75694569841777552706544696664, −2.57065998078734261502241065919, −2.04019124415443461737073634313, −1.61975033420697757987100896366, −1.60984616383238790659676246936, −1.46686180927517872610812203062, −0.52971129587738853294087149004, −0.51341977881179666706024373274, 0.51341977881179666706024373274, 0.52971129587738853294087149004, 1.46686180927517872610812203062, 1.60984616383238790659676246936, 1.61975033420697757987100896366, 2.04019124415443461737073634313, 2.57065998078734261502241065919, 2.75694569841777552706544696664, 2.79454867483433052481827070279, 3.22844972834352146454980873119, 3.28044049645570951462659754632, 3.51043025878399467315072527523, 3.99351048552983604914924096749, 4.02125708518221809874551826142, 4.27930073702111663760417087140, 4.66340665222888927928543807467, 4.75188689505607587174472311312, 4.75365138107859220109840666807, 5.40827160720220275282425380912, 5.55084746919041315040853094391, 5.56159385640590190717119270674, 5.81123908823528425677135855811, 6.20094730782170497679266898390, 6.38827513399849217724341860481, 6.41828526041226557928924594561

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