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Properties

Label 8-605e4-1.1-c1e4-0-11
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·4-s − 4·5-s − 3·7-s + 4·8-s − 6·9-s + 4·10-s − 13-s + 3·14-s + 8·16-s + 17-s + 6·18-s − 20·19-s + 16·20-s + 5·23-s + 10·25-s + 26-s − 5·27-s + 12·28-s − 12·29-s − 5·31-s − 5·32-s − 34-s + 12·35-s + 24·36-s + 7·37-s + 20·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 2·4-s − 1.78·5-s − 1.13·7-s + 1.41·8-s − 2·9-s + 1.26·10-s − 0.277·13-s + 0.801·14-s + 2·16-s + 0.242·17-s + 1.41·18-s − 4.58·19-s + 3.57·20-s + 1.04·23-s + 2·25-s + 0.196·26-s − 0.962·27-s + 2.26·28-s − 2.22·29-s − 0.898·31-s − 0.883·32-s − 0.171·34-s + 2.02·35-s + 4·36-s + 1.15·37-s + 3.24·38-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 5 T^{2} + 5 T^{3} + 13 T^{4} + 5 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} + 5 T^{3} + 17 T^{4} + 5 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 17 T^{2} + 40 T^{3} + 171 T^{4} + 40 p T^{5} + 17 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + T + 27 T^{2} + 32 T^{3} + 503 T^{4} + 32 p T^{5} + 27 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - T + 48 T^{2} - 19 T^{3} + 1073 T^{4} - 19 p T^{5} + 48 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 206 T^{2} + 1415 T^{3} + 7131 T^{4} + 1415 p T^{5} + 206 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 96 T^{2} - 335 T^{3} + 3347 T^{4} - 335 p T^{5} + 96 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 136 T^{2} + 873 T^{3} + 5755 T^{4} + 873 p T^{5} + 136 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 49 T^{2} + 340 T^{3} + 1741 T^{4} + 340 p T^{5} + 49 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 48 T^{2} + 49 T^{3} - 337 T^{4} + 49 p T^{5} + 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 170 T^{2} + 1179 T^{3} + 10259 T^{4} + 1179 p T^{5} + 170 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 293 T^{2} + 2740 T^{3} + 21711 T^{4} + 2740 p T^{5} + 293 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 167 T^{2} - 640 T^{3} + 11449 T^{4} - 640 p T^{5} + 167 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 169 T^{2} + 1438 T^{3} + 13237 T^{4} + 1438 p T^{5} + 169 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 163 T^{2} - 1044 T^{3} + 11443 T^{4} - 1044 p T^{5} + 163 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 267 T^{2} + 2118 T^{3} + 24963 T^{4} + 2118 p T^{5} + 267 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 290 T^{2} + 2805 T^{3} + 25803 T^{4} + 2805 p T^{5} + 290 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 238 T^{2} - 895 T^{3} + 23583 T^{4} - 895 p T^{5} + 238 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 302 T^{2} + 2397 T^{3} + 33423 T^{4} + 2397 p T^{5} + 302 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 34 T + 652 T^{2} + 8357 T^{3} + 83755 T^{4} + 8357 p T^{5} + 652 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 233 T^{2} - 1500 T^{3} + 23201 T^{4} - 1500 p T^{5} + 233 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 254 T^{2} + 1664 T^{3} + 31231 T^{4} + 1664 p T^{5} + 254 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 598 T^{2} - 8416 T^{3} + 94183 T^{4} - 8416 p T^{5} + 598 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.461448713343833532853397475397, −7.920841581654134894404328150083, −7.77042925084168389491816674765, −7.61792002822535937581635244342, −7.46936990535788958015923681226, −6.99950682901431182278685433213, −6.58072521883113617915473801395, −6.50125486045569515394467678516, −6.32686627799663483956854601499, −6.12026503765902627513151816615, −5.60767085312592818347142706152, −5.59591590602426816022072128566, −5.19052348752579281999339725195, −4.74965239119804516306382767148, −4.69280174077293820606452638905, −4.47680556368918909719046141740, −4.15454427297041553067919598302, −3.88150161957353591282259196853, −3.60592616505334352342799589288, −3.34177846670612872961654646283, −3.15263582214246796984365903508, −2.85273387277794221626883106963, −2.26687467016727898820150766433, −1.94021034518507580190273520563, −1.37217045192519409836679007673, 0, 0, 0, 0, 1.37217045192519409836679007673, 1.94021034518507580190273520563, 2.26687467016727898820150766433, 2.85273387277794221626883106963, 3.15263582214246796984365903508, 3.34177846670612872961654646283, 3.60592616505334352342799589288, 3.88150161957353591282259196853, 4.15454427297041553067919598302, 4.47680556368918909719046141740, 4.69280174077293820606452638905, 4.74965239119804516306382767148, 5.19052348752579281999339725195, 5.59591590602426816022072128566, 5.60767085312592818347142706152, 6.12026503765902627513151816615, 6.32686627799663483956854601499, 6.50125486045569515394467678516, 6.58072521883113617915473801395, 6.99950682901431182278685433213, 7.46936990535788958015923681226, 7.61792002822535937581635244342, 7.77042925084168389491816674765, 7.920841581654134894404328150083, 8.461448713343833532853397475397

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