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Properties

Label 8-6825e4-1.1-c1e4-0-8
Degree $8$
Conductor $2.170\times 10^{15}$
Sign $1$
Analytic cond. $8.82102\times 10^{6}$
Root an. cond. $7.38226$
Motivic weight $1$
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 + 4·3-s − 3·4-s − 8·6-s + 4·7-s + 10·8-s + 10·9-s − 4·11-s − 12·12-s + 4·13-s − 8·14-s − 2·17-s − 20·18-s − 2·19-s + 16·21-s + 8·22-s − 8·23-s + 40·24-s − 8·26-s + 20·27-s − 12·28-s − 6·29-s − 2·31-s − 18·32-s − 16·33-s + 4·34-s − 30·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s − 3/2·4-s − 3.26·6-s + 1.51·7-s + 3.53·8-s + 10/3·9-s − 1.20·11-s − 3.46·12-s + 1.10·13-s − 2.13·14-s − 0.485·17-s − 4.71·18-s − 0.458·19-s + 3.49·21-s + 1.70·22-s − 1.66·23-s + 8.16·24-s − 1.56·26-s + 3.84·27-s − 2.26·28-s − 1.11·29-s − 0.359·31-s − 3.18·32-s − 2.78·33-s + 0.685·34-s − 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8.82102\times 10^{6}\)
Root analytic conductor: \(7.38226\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
good2$D_{4}$ \( ( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 30 T^{2} + 96 T^{3} + 479 T^{4} + 96 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 42 T^{2} + 100 T^{3} + 951 T^{4} + 100 p T^{5} + 42 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 65 T^{2} + 82 T^{3} + 91 p T^{4} + 82 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 101 T^{2} + 534 T^{3} + 3569 T^{4} + 534 p T^{5} + 101 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 117 T^{2} + 518 T^{3} + 5105 T^{4} + 518 p T^{5} + 117 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 58 T^{2} + 264 T^{3} + 1655 T^{4} + 264 p T^{5} + 58 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 178 T^{2} + 1140 T^{3} + 10439 T^{4} + 1140 p T^{5} + 178 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 154 T^{2} + 9271 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 18 T + 191 T^{2} + 1664 T^{3} + 12489 T^{4} + 1664 p T^{5} + 191 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 222 T^{2} + 1640 T^{3} + 16211 T^{4} + 1640 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 202 T^{2} - 1580 T^{3} + 15799 T^{4} - 1580 p T^{5} + 202 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 187 T^{2} - 692 T^{3} + 15345 T^{4} - 692 p T^{5} + 187 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 374 T^{2} + 4000 T^{3} + 631 p T^{4} + 4000 p T^{5} + 374 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 158 T^{2} + 1640 T^{3} + 12499 T^{4} + 1640 p T^{5} + 158 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 255 T^{2} + 2216 T^{3} + 24209 T^{4} + 2216 p T^{5} + 255 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 318 T^{2} + 2980 T^{3} + 35571 T^{4} + 2980 p T^{5} + 318 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 26 T + 527 T^{2} + 6628 T^{3} + 70285 T^{4} + 6628 p T^{5} + 527 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 171 T^{2} + 1056 T^{3} + 10029 T^{4} + 1056 p T^{5} + 171 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 20 T + 431 T^{2} - 5380 T^{3} + 60781 T^{4} - 5380 p T^{5} + 431 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 449 T^{2} + 4682 T^{3} + 68209 T^{4} + 4682 p T^{5} + 449 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.81904268715816889705289093057, −5.60167487093016539901850019795, −5.53506889701874981277586950574, −5.32098034293969771216540902679, −5.30207257103660672211762758307, −4.91539856415968392392005953637, −4.71797232111226833278091529527, −4.53796219825313599915226422003, −4.45828412545879634768926638423, −4.13888377881462967193682836110, −4.11728572121890919687193097707, −3.92063264080310109328682357460, −3.86432656544619150302292634570, −3.33962141249644402522879060411, −3.30950470805136701548321595128, −3.01317181692908205255731873709, −2.97392361987606415362252658773, −2.46042816597050830490054002451, −2.40434199228520828984595146455, −2.07461106998119427887240828882, −1.89246378015758408954679526186, −1.40729324666876766715359273883, −1.37281039725513036497708692986, −1.36207860891611039286206709517, −1.22420973054299432978502676923, 0, 0, 0, 0, 1.22420973054299432978502676923, 1.36207860891611039286206709517, 1.37281039725513036497708692986, 1.40729324666876766715359273883, 1.89246378015758408954679526186, 2.07461106998119427887240828882, 2.40434199228520828984595146455, 2.46042816597050830490054002451, 2.97392361987606415362252658773, 3.01317181692908205255731873709, 3.30950470805136701548321595128, 3.33962141249644402522879060411, 3.86432656544619150302292634570, 3.92063264080310109328682357460, 4.11728572121890919687193097707, 4.13888377881462967193682836110, 4.45828412545879634768926638423, 4.53796219825313599915226422003, 4.71797232111226833278091529527, 4.91539856415968392392005953637, 5.30207257103660672211762758307, 5.32098034293969771216540902679, 5.53506889701874981277586950574, 5.60167487093016539901850019795, 5.81904268715816889705289093057

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