↑

Properties

Label 8-15e8-1.1-c11e4-0-3
Degree $8$
Conductor $2562890625$
Sign $1$
Analytic cond. $8.93204\times 10^{8}$
Root an. cond. $13.1482$
Motivic weight $11$
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  − 2.88e3·4-s + 1.23e6·11-s + 6.59e3·16-s − 1.06e7·19-s + 1.88e8·29-s + 4.89e8·31-s + 1.49e9·41-s − 3.56e9·44-s + 5.89e9·49-s + 1.46e10·59-s − 3.03e9·61-s + 1.17e10·64-s − 6.58e10·71-s + 3.06e10·76-s + 6.60e9·79-s − 2.53e10·89-s − 1.72e11·101-s + 3.24e10·109-s − 5.42e11·116-s + 6.56e11·121-s − 1.40e12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.40·4-s + 2.31·11-s + 0.00157·16-s − 0.986·19-s + 1.70·29-s + 3.06·31-s + 2.01·41-s − 3.25·44-s + 2.98·49-s + 2.66·59-s − 0.459·61-s + 1.37·64-s − 4.33·71-s + 1.38·76-s + 0.241·79-s − 0.481·89-s − 1.63·101-s + 0.201·109-s − 2.39·116-s + 2.29·121-s − 4.31·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8.93204\times 10^{8}\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(4.804020887\)
\(L(\frac12)\) \(\approx\) \(4.804020887\)
\(L(\frac{13}{2})\) 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 \( 1 \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 45 p^{6} T^{2} + 129497 p^{6} T^{4} + 45 p^{28} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 - 5896330900 T^{2} + 325445393122406502 p^{2} T^{4} - 5896330900 p^{22} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 - 618176 T + 245194892966 T^{2} - 618176 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 1140153047180 T^{2} + \)\(55\!\cdots\!38\)\( T^{4} - 1140153047180 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 - 116879825889660 T^{2} + \)\(57\!\cdots\!78\)\( T^{4} - 116879825889660 p^{22} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 + 280280 p T + 194538827137638 T^{2} + 280280 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 2072692881139220 T^{2} + \)\(28\!\cdots\!58\)\( T^{4} - 2072692881139220 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 - 3246220 p T + 23426350431097358 T^{2} - 3246220 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 244543464 T + 34393316207729486 T^{2} - 244543464 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 273812295186452780 T^{2} + \)\(81\!\cdots\!38\)\( T^{4} - 273812295186452780 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 - 745743316 T + 929792912462405846 T^{2} - 745743316 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3285052879347844100 T^{2} + \)\(43\!\cdots\!98\)\( T^{4} - 3285052879347844100 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 - 8397835342270441140 T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - 8397835342270441140 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 + 5703220206102687060 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + 5703220206102687060 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 - 7317515560 T + 55027608950440780118 T^{2} - 7317515560 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 1516425676 T + 85869525433683691566 T^{2} + 1516425676 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(60\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 + 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} + 32938471544 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!58\)\( T^{4} - \)\(66\!\cdots\!20\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 - 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} - 3302823120 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(64\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + 12674770860 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{22} T^{6} + p^{44} T^{8} \)
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

−6.91225852503154036294341562168, −6.65335780868797008196225426179, −6.59755964609177548297705814818, −6.05095093366536032080627795486, −5.84705348151016428413046841557, −5.78781116041827701055388396925, −5.45831539857748573303375691473, −4.92687550801472266179770464588, −4.50998906466315721764440868686, −4.44849592249475814622229958704, −4.28995871681384727409124845265, −4.23076729874191125543158815548, −3.99401561358937485172684093768, −3.47083626543787426310353834279, −3.11801442749107035509862255341, −2.88557754036166460222395055429, −2.59046360338140322427379292204, −2.23734418909473713907390015537, −2.03187065974828544409462811385, −1.54349951267763020048305665459, −1.21571555607082564101283858009, −0.887010004766636975769588915160, −0.77210174862532291434391745737, −0.65435434685962109062112904877, −0.21853291106537972863883562443, 0.21853291106537972863883562443, 0.65435434685962109062112904877, 0.77210174862532291434391745737, 0.887010004766636975769588915160, 1.21571555607082564101283858009, 1.54349951267763020048305665459, 2.03187065974828544409462811385, 2.23734418909473713907390015537, 2.59046360338140322427379292204, 2.88557754036166460222395055429, 3.11801442749107035509862255341, 3.47083626543787426310353834279, 3.99401561358937485172684093768, 4.23076729874191125543158815548, 4.28995871681384727409124845265, 4.44849592249475814622229958704, 4.50998906466315721764440868686, 4.92687550801472266179770464588, 5.45831539857748573303375691473, 5.78781116041827701055388396925, 5.84705348151016428413046841557, 6.05095093366536032080627795486, 6.59755964609177548297705814818, 6.65335780868797008196225426179, 6.91225852503154036294341562168

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