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Properties

Label 8-6825e4-1.1-c1e4-0-0
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 4·3-s + 3·4-s − 12·6-s + 4·7-s + 2·8-s + 10·9-s − 6·11-s − 12·12-s − 4·13-s + 12·14-s + 4·16-s + 6·17-s + 30·18-s − 7·19-s − 16·21-s − 18·22-s + 3·23-s − 8·24-s − 12·26-s − 20·27-s + 12·28-s − 5·29-s − 6·31-s + 4·32-s + 24·33-s + 18·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 2.30·3-s + 3/2·4-s − 4.89·6-s + 1.51·7-s + 0.707·8-s + 10/3·9-s − 1.80·11-s − 3.46·12-s − 1.10·13-s + 3.20·14-s + 16-s + 1.45·17-s + 7.07·18-s − 1.60·19-s − 3.49·21-s − 3.83·22-s + 0.625·23-s − 1.63·24-s − 2.35·26-s − 3.84·27-s + 2.26·28-s − 0.928·29-s − 1.07·31-s + 0.707·32-s + 4.17·33-s + 3.08·34-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: \(0\)
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

\(L(1)\) \(\approx\) \(4.984333010\)
\(L(\frac12)\) \(\approx\) \(4.984333010\)
\(L(\frac{3}{2})\) 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$C_4\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 11 T^{3} + 17 T^{4} - 11 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 41 T^{2} + 142 T^{3} + 596 T^{4} + 142 p T^{5} + 41 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 65 T^{2} - 290 T^{3} + 1628 T^{4} - 290 p T^{5} + 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 51 T^{2} + 263 T^{3} + 80 p T^{4} + 263 p T^{5} + 51 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 3 T - 5 T^{2} + 45 T^{3} + 328 T^{4} + 45 p T^{5} - 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 81 T^{2} + 15 p T^{3} + 3116 T^{4} + 15 p^{2} T^{5} + 81 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 121 T^{2} + 502 T^{3} + 5516 T^{4} + 502 p T^{5} + 121 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 31 T^{2} - 76 T^{3} + 1072 T^{4} - 76 p T^{5} + 31 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 221 T^{2} + 1758 T^{3} + 14636 T^{4} + 1758 p T^{5} + 221 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 155 T^{2} - 1285 T^{3} + 10128 T^{4} - 1285 p T^{5} + 155 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 165 T^{2} - 30 p T^{3} + 11588 T^{4} - 30 p^{2} T^{5} + 165 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 307 T^{2} - 3280 T^{3} + 26824 T^{4} - 3280 p T^{5} + 307 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 85 T^{2} + 1073 T^{3} + 12204 T^{4} + 1073 p T^{5} + 85 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 149 T^{2} - 210 T^{3} + 12420 T^{4} - 210 p T^{5} + 149 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 433 T^{2} - 4886 T^{3} + 49348 T^{4} - 4886 p T^{5} + 433 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + T + 209 T^{2} + 341 T^{3} + 19900 T^{4} + 341 p T^{5} + 209 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 121 T^{2} + 330 T^{3} + 5252 T^{4} + 330 p T^{5} + 121 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 201 T^{2} + 2205 T^{3} + 19340 T^{4} + 2205 p T^{5} + 201 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 25 T + 463 T^{2} - 6005 T^{3} + 61144 T^{4} - 6005 p T^{5} + 463 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 287 T^{2} + 775 T^{3} + 34168 T^{4} + 775 p T^{5} + 287 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 271 T^{2} + 1949 T^{3} + 33832 T^{4} + 1949 p T^{5} + 271 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.52171529626063878155843689430, −5.31221535406385757212837030437, −5.25951917376572499046665088516, −5.07618940836813943237069043797, −4.96741261385326917402884158610, −4.68202380550908197581065086184, −4.46438472435408185781942741108, −4.45160778229154431790954039403, −4.36028312080852787764773012830, −3.88121554777754315851637013715, −3.87886488882076330276986950342, −3.71506333878880505278755473626, −3.67601777726795647333661977866, −2.96935610403698440802333908488, −2.91770934134385228498339730553, −2.76375571286063335557185993123, −2.29413222896496298889497047697, −2.17628683768085206158426367665, −2.08994944771827775725473728055, −1.70049233051665410710628137663, −1.50508307241192662839333851454, −1.12245981976128372911934263655, −0.70518858670219237193679628613, −0.65205712984631581296177674653, −0.27150361584375445472246052239, 0.27150361584375445472246052239, 0.65205712984631581296177674653, 0.70518858670219237193679628613, 1.12245981976128372911934263655, 1.50508307241192662839333851454, 1.70049233051665410710628137663, 2.08994944771827775725473728055, 2.17628683768085206158426367665, 2.29413222896496298889497047697, 2.76375571286063335557185993123, 2.91770934134385228498339730553, 2.96935610403698440802333908488, 3.67601777726795647333661977866, 3.71506333878880505278755473626, 3.87886488882076330276986950342, 3.88121554777754315851637013715, 4.36028312080852787764773012830, 4.45160778229154431790954039403, 4.46438472435408185781942741108, 4.68202380550908197581065086184, 4.96741261385326917402884158610, 5.07618940836813943237069043797, 5.25951917376572499046665088516, 5.31221535406385757212837030437, 5.52171529626063878155843689430

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