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Properties

Label 8-39e8-1.1-c3e4-0-4
Degree $8$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $6.48606\times 10^{7}$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·4-s + 6·5-s − 14·7-s − 4·8-s − 12·10-s − 40·11-s + 28·14-s + 15·16-s − 98·17-s + 124·19-s − 18·20-s + 80·22-s − 104·23-s − 261·25-s + 42·28-s − 194·29-s + 26·31-s + 2·32-s + 196·34-s − 84·35-s + 102·37-s − 248·38-s − 24·40-s + 1.05e3·41-s + 450·43-s + 120·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 3/8·4-s + 0.536·5-s − 0.755·7-s − 0.176·8-s − 0.379·10-s − 1.09·11-s + 0.534·14-s + 0.234·16-s − 1.39·17-s + 1.49·19-s − 0.201·20-s + 0.775·22-s − 0.942·23-s − 2.08·25-s + 0.283·28-s − 1.24·29-s + 0.150·31-s + 0.0110·32-s + 0.988·34-s − 0.405·35-s + 0.453·37-s − 1.05·38-s − 0.0948·40-s + 4.01·41-s + 1.59·43-s + 0.411·44-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(6.48606\times 10^{7}\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + p T + 7 T^{2} + 3 p^{3} T^{3} + 31 p T^{4} + 3 p^{6} T^{5} + 7 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 6 T + 297 T^{2} - 2094 T^{3} + 49084 T^{4} - 2094 p^{3} T^{5} + 297 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 2 p T + 249 T^{2} + 1754 T^{3} + 141728 T^{4} + 1754 p^{3} T^{5} + 249 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 40 T + 4068 T^{2} + 116616 T^{3} + 7313302 T^{4} + 116616 p^{3} T^{5} + 4068 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 98 T + 12397 T^{2} + 986190 T^{3} + 96109736 T^{4} + 986190 p^{3} T^{5} + 12397 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 124 T + 21504 T^{2} - 1798684 T^{3} + 202620494 T^{4} - 1798684 p^{3} T^{5} + 21504 p^{6} T^{6} - 124 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 104 T + 41844 T^{2} + 3340584 T^{3} + 719588614 T^{4} + 3340584 p^{3} T^{5} + 41844 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 194 T + 64761 T^{2} + 7214754 T^{3} + 1694674348 T^{4} + 7214754 p^{3} T^{5} + 64761 p^{6} T^{6} + 194 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 26 T + 38189 T^{2} + 5395494 T^{3} + 828557428 T^{4} + 5395494 p^{3} T^{5} + 38189 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 102 T + 193281 T^{2} - 14822902 T^{3} + 14476248876 T^{4} - 14822902 p^{3} T^{5} + 193281 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 1054 T + 633469 T^{2} - 261177138 T^{3} + 78839658968 T^{4} - 261177138 p^{3} T^{5} + 633469 p^{6} T^{6} - 1054 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 450 T + 276753 T^{2} - 75560470 T^{3} + 29002070616 T^{4} - 75560470 p^{3} T^{5} + 276753 p^{6} T^{6} - 450 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 96 T - 19308 T^{2} - 124896 T^{3} + 17303792422 T^{4} - 124896 p^{3} T^{5} - 19308 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 262 T + 483789 T^{2} + 119532570 T^{3} + 100466115904 T^{4} + 119532570 p^{3} T^{5} + 483789 p^{6} T^{6} + 262 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 308 T + 557680 T^{2} + 98956308 T^{3} + 142593856142 T^{4} + 98956308 p^{3} T^{5} + 557680 p^{6} T^{6} + 308 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 928 T + 1041074 T^{2} + 583837824 T^{3} + 364336742755 T^{4} + 583837824 p^{3} T^{5} + 1041074 p^{6} T^{6} + 928 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1134 T + 1218801 T^{2} + 918506522 T^{3} + 562458950376 T^{4} + 918506522 p^{3} T^{5} + 1218801 p^{6} T^{6} + 1134 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1064 T + 1201044 T^{2} + 690200040 T^{3} + 504474394630 T^{4} + 690200040 p^{3} T^{5} + 1201044 p^{6} T^{6} + 1064 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 952 T + 768042 T^{2} - 3278032 p T^{3} + 174760946603 T^{4} - 3278032 p^{4} T^{5} + 768042 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 746 T + 2156493 T^{2} + 1122200666 T^{3} + 1640976331028 T^{4} + 1122200666 p^{3} T^{5} + 2156493 p^{6} T^{6} + 746 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 404 T + 21824 p T^{2} - 594579060 T^{3} + 1476544322126 T^{4} - 594579060 p^{3} T^{5} + 21824 p^{7} T^{6} - 404 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1620 T + 2005896 T^{2} + 1665077532 T^{3} + 1565610279790 T^{4} + 1665077532 p^{3} T^{5} + 2005896 p^{6} T^{6} + 1620 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2166 T + 4160745 T^{2} - 5948172454 T^{3} + 5928410141748 T^{4} - 5948172454 p^{3} T^{5} + 4160745 p^{6} T^{6} - 2166 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.70572028554428766110033985005, −6.49952571990433946354298796322, −6.36765556057263244315687460487, −5.96439711141188480340226964364, −5.89655702028838734401575589108, −5.86996071652431033583379712785, −5.61310940247907610774611167648, −5.45190213179498770103211177960, −5.15345175896201926381732051056, −4.67342567910907310653353159670, −4.60797453178107970733814881625, −4.28423118064832192250339290215, −4.27954986441463774820334998609, −3.72903590617930109952302730860, −3.69946158314778940149507540391, −3.58939149680003314698650400328, −2.85072091404331518070366828806, −2.72585200419849673002162077234, −2.71773770013627145978133151951, −2.43724007709548561130663359666, −2.14227727077294197855542567236, −1.75649509311200649094213002396, −1.30028219501255062181861427574, −1.23411088150481203830956589587, −0.852543089929135927742108642246, 0, 0, 0, 0, 0.852543089929135927742108642246, 1.23411088150481203830956589587, 1.30028219501255062181861427574, 1.75649509311200649094213002396, 2.14227727077294197855542567236, 2.43724007709548561130663359666, 2.71773770013627145978133151951, 2.72585200419849673002162077234, 2.85072091404331518070366828806, 3.58939149680003314698650400328, 3.69946158314778940149507540391, 3.72903590617930109952302730860, 4.27954986441463774820334998609, 4.28423118064832192250339290215, 4.60797453178107970733814881625, 4.67342567910907310653353159670, 5.15345175896201926381732051056, 5.45190213179498770103211177960, 5.61310940247907610774611167648, 5.86996071652431033583379712785, 5.89655702028838734401575589108, 5.96439711141188480340226964364, 6.36765556057263244315687460487, 6.49952571990433946354298796322, 6.70572028554428766110033985005

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