↑

Properties

Label 8-39e8-1.1-c3e4-0-6
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

Downloads

Learn more

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$C_2 \wr S_4$ \( 1 + 2166 T + 4160745 T^{2} + 5948172454 T^{3} + 5928410141748 T^{4} + 5948172454 p^{3} T^{5} + 4160745 p^{6} T^{6} + 2166 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.81063841629145540808180810489, −6.63724092095745595669090911253, −6.29053387196781046624742943523, −6.25857306493786151174289933495, −5.89486899117122821424465377837, −5.64486075152092329768630900112, −5.31553799592030991497642614208, −5.30884709906757649436372859700, −5.12211299158593502553907042954, −4.65266403210223881873987067870, −4.50261261544489429955864506275, −4.27943795959935446084970177265, −4.26901241580122996639616960403, −3.88360343977545662423930801983, −3.79342937202612114688722291303, −3.65876349281430030392574919349, −3.28728943102379731908203741438, −3.02665424742928672294435278308, −2.43849276021562390482553847028, −2.35033188474402870840584206709, −2.02475394237595271426670357684, −1.80842030511212146768150094984, −1.63833666428428244574655401783, −1.14076464411841099358431151573, −1.12927364861316203486330634664, 0, 0, 0, 0, 1.12927364861316203486330634664, 1.14076464411841099358431151573, 1.63833666428428244574655401783, 1.80842030511212146768150094984, 2.02475394237595271426670357684, 2.35033188474402870840584206709, 2.43849276021562390482553847028, 3.02665424742928672294435278308, 3.28728943102379731908203741438, 3.65876349281430030392574919349, 3.79342937202612114688722291303, 3.88360343977545662423930801983, 4.26901241580122996639616960403, 4.27943795959935446084970177265, 4.50261261544489429955864506275, 4.65266403210223881873987067870, 5.12211299158593502553907042954, 5.30884709906757649436372859700, 5.31553799592030991497642614208, 5.64486075152092329768630900112, 5.89486899117122821424465377837, 6.25857306493786151174289933495, 6.29053387196781046624742943523, 6.63724092095745595669090911253, 6.81063841629145540808180810489

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