↑

Properties

Label 8-45e8-1.1-c1e4-0-12
Degree $8$
Conductor $1.682\times 10^{13}$
Sign $1$
Analytic cond. $68361.0$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7-s + 8-s − 11-s − 2·13-s − 2·14-s + 16-s − 11·17-s + 2·19-s + 2·22-s − 15·23-s + 4·26-s + 29-s − 4·31-s + 2·32-s + 22·34-s + 37-s − 4·38-s − 5·41-s + 10·43-s + 30·46-s − 20·47-s − 15·49-s − 20·53-s + 56-s − 2·58-s + 17·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 2.66·17-s + 0.458·19-s + 0.426·22-s − 3.12·23-s + 0.784·26-s + 0.185·29-s − 0.718·31-s + 0.353·32-s + 3.77·34-s + 0.164·37-s − 0.648·38-s − 0.780·41-s + 1.52·43-s + 4.42·46-s − 2.91·47-s − 2.14·49-s − 2.74·53-s + 0.133·56-s − 0.262·58-s + 2.21·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(68361.0\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + 7 T^{3} + 11 T^{4} + 7 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T + 16 T^{2} + 3 T^{3} + 117 T^{4} + 3 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + T + 19 T^{2} + 74 T^{3} + 167 T^{4} + 74 p T^{5} + 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 22 T^{2} + 73 T^{3} + 341 T^{4} + 73 p T^{5} + 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 88 T^{2} + 451 T^{3} + 2111 T^{4} + 451 p T^{5} + 88 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 34 p T^{5} + 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 149 T^{2} + 1026 T^{3} + 5553 T^{4} + 1026 p T^{5} + 149 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - T + 76 T^{2} + 55 T^{3} + 2597 T^{4} + 55 p T^{5} + 76 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 82 T^{2} + 345 T^{3} + 3405 T^{4} + 345 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - T + 49 T^{2} + 392 T^{3} + 241 T^{4} + 392 p T^{5} + 49 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 139 T^{2} + 454 T^{3} + 7829 T^{4} + 454 p T^{5} + 139 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 124 T^{2} - 704 T^{3} + 6293 T^{4} - 704 p T^{5} + 124 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 295 T^{2} + 2890 T^{3} + 22931 T^{4} + 2890 p T^{5} + 295 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 298 T^{2} + 3001 T^{3} + 25499 T^{4} + 3001 p T^{5} + 298 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 238 T^{2} - 2095 T^{3} + 18809 T^{4} - 2095 p T^{5} + 238 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 241 T^{2} + 2288 T^{3} + 21959 T^{4} + 2288 p T^{5} + 241 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 304 T^{2} + 3117 T^{3} + 32001 T^{4} + 3117 p T^{5} + 304 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 1441 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 196 T^{2} + 679 T^{3} + 18071 T^{4} + 679 p T^{5} + 196 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 283 T^{2} + 1590 T^{3} + 32439 T^{4} + 1590 p T^{5} + 283 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 620 T^{2} + 8415 T^{3} + 89871 T^{4} + 8415 p T^{5} + 620 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 257 T^{2} + 1998 T^{3} + 31929 T^{4} + 1998 p T^{5} + 257 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 469 T^{2} - 5368 T^{3} + 71215 T^{4} - 5368 p T^{5} + 469 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.89760549371695437866898832037, −6.70700497581541290936365073488, −6.47334333638052078220863662945, −6.38129346984140351166601730037, −6.09289230354130617016501313484, −5.95140714846136624543712067366, −5.86568239995300580884685742669, −5.34812065798393461988155422626, −5.27732118715960555456748447511, −4.91299008063205079723690735117, −4.80171131879282642587045653597, −4.50996240955195850604901440314, −4.41201842595078691525692038900, −4.24033210769337804892263384864, −3.84118937756002627122721746976, −3.74062848868596326630491864909, −3.42533733961646664872878720009, −2.93749776227674429290200421430, −2.85514867383312981471647511934, −2.64766884291166474693644613359, −2.19126052475320047544388723180, −2.05447917118210269183584522608, −1.61233063490130091502685756926, −1.45607740332670872056835916814, −1.24517492062830353832938267944, 0, 0, 0, 0, 1.24517492062830353832938267944, 1.45607740332670872056835916814, 1.61233063490130091502685756926, 2.05447917118210269183584522608, 2.19126052475320047544388723180, 2.64766884291166474693644613359, 2.85514867383312981471647511934, 2.93749776227674429290200421430, 3.42533733961646664872878720009, 3.74062848868596326630491864909, 3.84118937756002627122721746976, 4.24033210769337804892263384864, 4.41201842595078691525692038900, 4.50996240955195850604901440314, 4.80171131879282642587045653597, 4.91299008063205079723690735117, 5.27732118715960555456748447511, 5.34812065798393461988155422626, 5.86568239995300580884685742669, 5.95140714846136624543712067366, 6.09289230354130617016501313484, 6.38129346984140351166601730037, 6.47334333638052078220863662945, 6.70700497581541290936365073488, 6.89760549371695437866898832037

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