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Properties

Label 8-1875e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.236\times 10^{13}$
Sign $1$
Analytic cond. $50247.3$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s + 2·4-s + 8·6-s + 2·7-s + 5·8-s + 10·9-s − 7·11-s + 8·12-s + 13-s + 4·14-s + 5·16-s + 2·17-s + 20·18-s + 5·19-s + 8·21-s − 14·22-s + 23-s + 20·24-s + 2·26-s + 20·27-s + 4·28-s − 20·29-s + 23·31-s − 2·32-s − 28·33-s + 4·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 3.26·6-s + 0.755·7-s + 1.76·8-s + 10/3·9-s − 2.11·11-s + 2.30·12-s + 0.277·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s + 4.71·18-s + 1.14·19-s + 1.74·21-s − 2.98·22-s + 0.208·23-s + 4.08·24-s + 0.392·26-s + 3.84·27-s + 0.755·28-s − 3.71·29-s + 4.13·31-s − 0.353·32-s − 4.87·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(50247.3\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(35.63305874\)
\(L(\frac12)\) \(\approx\) \(35.63305874\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2^2:C_4$ \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 12 T^{2} - 15 T^{3} + 61 T^{4} - 15 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 38 T^{2} + 139 T^{3} + 485 T^{4} + 139 p T^{5} + 38 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - T + 38 T^{2} - 15 T^{3} + 641 T^{4} - 15 p T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 62 T^{2} - 95 T^{3} + 1541 T^{4} - 95 p T^{5} + 62 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 41 T^{2} - 210 T^{3} + 1111 T^{4} - 210 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - T + 68 T^{2} - 5 p T^{3} + 2051 T^{4} - 5 p^{2} T^{5} + 68 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 256 T^{2} + 2145 T^{3} + 13571 T^{4} + 2145 p T^{5} + 256 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 23 T + 308 T^{2} - 2751 T^{3} + 17885 T^{4} - 2751 p T^{5} + 308 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 102 T^{2} - 175 T^{3} + 4811 T^{4} - 175 p T^{5} + 102 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 233 T^{2} - 2040 T^{3} + 16241 T^{4} - 2040 p T^{5} + 233 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 152 T^{2} - 245 T^{3} + 10181 T^{4} - 245 p T^{5} + 152 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 178 T^{2} + 585 T^{3} + 13511 T^{4} + 585 p T^{5} + 178 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 191 T^{2} + 1440 T^{3} + 11931 T^{4} + 1440 p T^{5} + 191 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 188 T^{2} + 444 T^{3} + 15485 T^{4} + 444 p T^{5} + 188 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 237 T^{2} - 390 T^{3} + 22951 T^{4} - 390 p T^{5} + 237 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 68 T^{2} - 811 T^{3} + 485 T^{4} - 811 p T^{5} + 68 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 243 T^{2} - 2010 T^{3} + 19951 T^{4} - 2010 p T^{5} + 243 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 35 T + 636 T^{2} - 7945 T^{3} + 78161 T^{4} - 7945 p T^{5} + 636 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 278 T^{2} - 2680 T^{3} + 31391 T^{4} - 2680 p T^{5} + 278 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 35 T + 776 T^{2} + 11375 T^{3} + 125591 T^{4} + 11375 p T^{5} + 776 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 302 T^{2} - 2760 T^{3} + 41871 T^{4} - 2760 p T^{5} + 302 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.61105896077978953567954110261, −6.27249498231025184251109000148, −6.26097941179739414795313788090, −5.81740036152896033280465309952, −5.52360225195437263871588884970, −5.31313643518395762088516634701, −5.19556974659761105682144448394, −5.03729246103247927089008975375, −4.84035549041700819034464343803, −4.44071315638205817433866103979, −4.31306932818624656674694980169, −4.30036636280699055946058168804, −4.03607478058044305324747557416, −3.51890248366556082225406699454, −3.39093853213543805150769537653, −3.24397678861649761029938054500, −3.10169617875115115336089245772, −2.67251453519567784684462333184, −2.52322699462522008965164135012, −2.17189411661529837186689437601, −2.01194103234678130750228730253, −1.71991579970782834998441634687, −1.53805828270281050196862763619, −0.913290204547767087296950666723, −0.61628103870051830695030692803, 0.61628103870051830695030692803, 0.913290204547767087296950666723, 1.53805828270281050196862763619, 1.71991579970782834998441634687, 2.01194103234678130750228730253, 2.17189411661529837186689437601, 2.52322699462522008965164135012, 2.67251453519567784684462333184, 3.10169617875115115336089245772, 3.24397678861649761029938054500, 3.39093853213543805150769537653, 3.51890248366556082225406699454, 4.03607478058044305324747557416, 4.30036636280699055946058168804, 4.31306932818624656674694980169, 4.44071315638205817433866103979, 4.84035549041700819034464343803, 5.03729246103247927089008975375, 5.19556974659761105682144448394, 5.31313643518395762088516634701, 5.52360225195437263871588884970, 5.81740036152896033280465309952, 6.26097941179739414795313788090, 6.27249498231025184251109000148, 6.61105896077978953567954110261

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