↑

Properties

Label 8-6762e4-1.1-c1e4-0-5
Degree $8$
Conductor $2.091\times 10^{15}$
Sign $1$
Analytic cond. $8.49980\times 10^{6}$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 4·3-s + 10·4-s − 2·5-s − 16·6-s − 20·8-s + 10·9-s + 8·10-s + 6·11-s + 40·12-s + 2·13-s − 8·15-s + 35·16-s + 2·17-s − 40·18-s + 6·19-s − 20·20-s − 24·22-s + 4·23-s − 80·24-s − 5·25-s − 8·26-s + 20·27-s − 10·29-s + 32·30-s + 14·31-s − 56·32-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s − 0.894·5-s − 6.53·6-s − 7.07·8-s + 10/3·9-s + 2.52·10-s + 1.80·11-s + 11.5·12-s + 0.554·13-s − 2.06·15-s + 35/4·16-s + 0.485·17-s − 9.42·18-s + 1.37·19-s − 4.47·20-s − 5.11·22-s + 0.834·23-s − 16.3·24-s − 25-s − 1.56·26-s + 3.84·27-s − 1.85·29-s + 5.84·30-s + 2.51·31-s − 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(8.49980\times 10^{6}\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.891343206\)
\(L(\frac12)\) \(\approx\) \(9.891343206\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 9 T^{2} + 18 T^{3} + 48 T^{4} + 18 p T^{5} + 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 2 p T^{2} - 28 T^{3} + 43 T^{4} - 28 p T^{5} + 2 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 18 T^{2} + 34 T^{3} + 74 T^{4} + 34 p T^{5} + 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 43 T^{2} - 20 T^{3} + 822 T^{4} - 20 p T^{5} + 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 42 T^{2} - 262 T^{3} + 1170 T^{4} - 262 p T^{5} + 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 106 T^{2} + 744 T^{3} + 4655 T^{4} + 744 p T^{5} + 106 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 129 T^{2} - 882 T^{3} + 5440 T^{4} - 882 p T^{5} + 129 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 126 T^{2} - 650 T^{3} + 6618 T^{4} - 650 p T^{5} + 126 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 154 T^{2} + 1006 T^{3} + 9210 T^{4} + 1006 p T^{5} + 154 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 109 T^{2} - 358 T^{3} + 3240 T^{4} - 358 p T^{5} + 109 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 213 T^{2} - 930 T^{3} + 16948 T^{4} - 930 p T^{5} + 213 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 333 T^{2} + 3096 T^{3} + 25228 T^{4} + 3096 p T^{5} + 333 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 8 p T^{2} - 5796 T^{3} + 52222 T^{4} - 5796 p T^{5} + 8 p^{3} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 420 T^{2} - 4396 T^{3} + 38294 T^{4} - 4396 p T^{5} + 420 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 351 T^{2} - 3474 T^{3} + 39778 T^{4} - 3474 p T^{5} + 351 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 189 T^{2} + 446 T^{3} + 15312 T^{4} + 446 p T^{5} + 189 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 90 T^{2} - 296 T^{3} + 2951 T^{4} - 296 p T^{5} + 90 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 289 T^{2} + 2310 T^{3} + 30488 T^{4} + 2310 p T^{5} + 289 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 270 T^{2} - 2394 T^{3} + 29146 T^{4} - 2394 p T^{5} + 270 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 69 T^{2} - 112 T^{3} + 18456 T^{4} - 112 p T^{5} + 69 p^{2} T^{6} + p^{4} T^{8} \)
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

−5.82068442612804653889925802764, −5.38780369094117377459903581842, −5.26884156034182528557323251494, −5.06791382223919353499597590054, −4.96701490408086046290068105802, −4.28896861279596745583501426431, −4.27568355037334585043371680827, −4.25306216475229574238144773773, −3.95531719288277498878653965864, −3.65775622366152134404263224699, −3.49631966645590463413537630106, −3.43667702882843657373374427419, −3.31531178647946586316645768838, −2.89562905135828516371108811286, −2.69912255389558746729080503958, −2.67165667426232783471788202750, −2.33315765694758981814055319851, −1.95637716569164910594179834153, −1.78098661284018560053393906865, −1.74886349546506923038281153338, −1.56043011752996733796315935010, −0.902833255693116305210684387818, −0.76341662726038170841094006454, −0.69685823208087319735685100590, −0.69223071023237802667846257743, 0.69223071023237802667846257743, 0.69685823208087319735685100590, 0.76341662726038170841094006454, 0.902833255693116305210684387818, 1.56043011752996733796315935010, 1.74886349546506923038281153338, 1.78098661284018560053393906865, 1.95637716569164910594179834153, 2.33315765694758981814055319851, 2.67165667426232783471788202750, 2.69912255389558746729080503958, 2.89562905135828516371108811286, 3.31531178647946586316645768838, 3.43667702882843657373374427419, 3.49631966645590463413537630106, 3.65775622366152134404263224699, 3.95531719288277498878653965864, 4.25306216475229574238144773773, 4.27568355037334585043371680827, 4.28896861279596745583501426431, 4.96701490408086046290068105802, 5.06791382223919353499597590054, 5.26884156034182528557323251494, 5.38780369094117377459903581842, 5.82068442612804653889925802764

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