Properties

Label 8-546e4-1.1-c1e4-0-3
Degree $8$
Conductor $88873149456$
Sign $1$
Analytic cond. $361.309$
Root an. cond. $2.08802$
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 + 4-s + 9-s − 2·12-s + 6·17-s + 12·19-s + 8·23-s − 4·25-s + 2·27-s − 16·29-s + 36-s − 24·37-s − 24·41-s + 2·43-s + 49-s − 12·51-s − 40·53-s − 24·57-s + 6·59-s − 64-s + 18·67-s + 6·68-s − 16·69-s + 24·71-s + 8·75-s + 12·76-s − 24·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 1.45·17-s + 2.75·19-s + 1.66·23-s − 4/5·25-s + 0.384·27-s − 2.97·29-s + 1/6·36-s − 3.94·37-s − 3.74·41-s + 0.304·43-s + 1/7·49-s − 1.68·51-s − 5.49·53-s − 3.17·57-s + 0.781·59-s − 1/8·64-s + 2.19·67-s + 0.727·68-s − 1.92·69-s + 2.84·71-s + 0.923·75-s + 1.37·76-s − 2.70·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(361.309\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7215346244\)
\(L(\frac12)\) \(\approx\) \(0.7215346244\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 6 T + 5 T^{2} + 18 T^{3} + 60 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1707 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 8 T + 5 T^{2} - 104 T^{3} + 1480 T^{4} - 104 p T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 24 T + 310 T^{2} + 2832 T^{3} + 19659 T^{4} + 2832 p T^{5} + 310 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 2 T - 71 T^{2} + 22 T^{3} + 3604 T^{4} + 22 p T^{5} - 71 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 132 T^{2} + 8006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 20 T + 203 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 6 T + 69 T^{2} - 342 T^{3} + 476 T^{4} - 342 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 18 T + 265 T^{2} - 2826 T^{3} + 27396 T^{4} - 2826 p T^{5} + 265 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 24 T + 357 T^{2} - 3960 T^{3} + 35816 T^{4} - 3960 p T^{5} + 357 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 164 T^{2} + 14310 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 163 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 6 T + 93 T^{2} + 486 T^{3} - 292 T^{4} + 486 p T^{5} + 93 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 36 T + 730 T^{2} - 10728 T^{3} + 121299 T^{4} - 10728 p T^{5} + 730 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.69142391746753347386998246056, −7.56608541158320025189730485218, −7.09334399883042678238062887757, −6.93224731100799593607005033012, −6.91557202736186785793798584034, −6.89939411990615936289009071744, −6.29043475376490839500308371656, −5.95253303556012599295410803858, −5.89574126155553846202212122519, −5.34705187045503085558831304047, −5.33227071454830488358601285270, −5.26073625750644198636961068661, −4.95476999836256925769902203774, −4.88049162606403145839739213054, −4.31654837945409076956688125335, −3.61383730384312203796128503229, −3.58100304210979593434574221023, −3.40070455090560381716613569849, −3.14827916740683880830491384810, −3.04019460697396748757509041738, −2.07883979412458789837696819307, −1.90383509974229487257351589042, −1.45735749955846818859094979571, −1.26226376673888705563999383917, −0.29548496073519443075110509941, 0.29548496073519443075110509941, 1.26226376673888705563999383917, 1.45735749955846818859094979571, 1.90383509974229487257351589042, 2.07883979412458789837696819307, 3.04019460697396748757509041738, 3.14827916740683880830491384810, 3.40070455090560381716613569849, 3.58100304210979593434574221023, 3.61383730384312203796128503229, 4.31654837945409076956688125335, 4.88049162606403145839739213054, 4.95476999836256925769902203774, 5.26073625750644198636961068661, 5.33227071454830488358601285270, 5.34705187045503085558831304047, 5.89574126155553846202212122519, 5.95253303556012599295410803858, 6.29043475376490839500308371656, 6.89939411990615936289009071744, 6.91557202736186785793798584034, 6.93224731100799593607005033012, 7.09334399883042678238062887757, 7.56608541158320025189730485218, 7.69142391746753347386998246056

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