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Properties

Label 8-3640e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.756\times 10^{14}$
Sign $1$
Analytic cond. $713697.$
Root an. cond. $5.39124$
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  − 3-s + 4·5-s − 4·7-s − 6·9-s − 4·11-s + 4·13-s − 4·15-s − 17-s − 7·19-s + 4·21-s − 6·23-s + 10·25-s + 6·27-s + 29-s − 11·31-s + 4·33-s − 16·35-s − 3·37-s − 4·39-s − 5·41-s − 6·43-s − 24·45-s − 6·47-s + 10·49-s + 51-s − 2·53-s − 16·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 1.51·7-s − 2·9-s − 1.20·11-s + 1.10·13-s − 1.03·15-s − 0.242·17-s − 1.60·19-s + 0.872·21-s − 1.25·23-s + 2·25-s + 1.15·27-s + 0.185·29-s − 1.97·31-s + 0.696·33-s − 2.70·35-s − 0.493·37-s − 0.640·39-s − 0.780·41-s − 0.914·43-s − 3.57·45-s − 0.875·47-s + 10/7·49-s + 0.140·51-s − 0.274·53-s − 2.15·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(713697.\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + T + 7 T^{2} + 7 T^{3} + 28 T^{4} + 7 p T^{5} + 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + T + 53 T^{2} + 59 T^{3} + 1228 T^{4} + 59 p T^{5} + 53 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 71 T^{2} + 373 T^{3} + 1980 T^{4} + 373 p T^{5} + 71 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 72 T^{2} + 382 T^{3} + 2318 T^{4} + 382 p T^{5} + 72 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - T + 63 T^{2} + 149 T^{3} + 1688 T^{4} + 149 p T^{5} + 63 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 101 T^{2} + 437 T^{3} + 2736 T^{4} + 437 p T^{5} + 101 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 141 T^{2} + 321 T^{3} + 7692 T^{4} + 321 p T^{5} + 141 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 135 T^{2} + 575 T^{3} + 7832 T^{4} + 575 p T^{5} + 135 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 164 T^{2} + 718 T^{3} + 10422 T^{4} + 718 p T^{5} + 164 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 48 T^{2} + 14 T^{3} + 1118 T^{4} + 14 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 180 T^{2} + 230 T^{3} + 13478 T^{4} + 230 p T^{5} + 180 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + T + 17 T^{2} - 17 T^{3} + 5680 T^{4} - 17 p T^{5} + 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 224 T^{2} + 684 T^{3} + 19950 T^{4} + 684 p T^{5} + 224 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 259 T^{2} + 2167 T^{3} + 25744 T^{4} + 2167 p T^{5} + 259 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 344 T^{2} + 3456 T^{3} + 38750 T^{4} + 3456 p T^{5} + 344 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 196 T^{2} - 366 T^{3} + 20054 T^{4} - 366 p T^{5} + 196 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 217 T^{2} + 1655 T^{3} + 16028 T^{4} + 1655 p T^{5} + 217 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 216 T^{2} + 306 T^{3} + 23294 T^{4} + 306 p T^{5} + 216 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 95 T^{2} + 153 T^{3} + 7224 T^{4} + 153 p T^{5} + 95 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 328 T^{2} - 2344 T^{3} + 44878 T^{4} - 2344 p T^{5} + 328 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.36266633574154458096594057647, −6.03640927285934350426535462415, −6.03588414994209172792264608302, −5.86300646286894244226081144635, −5.85383996586136146947694339092, −5.36766461937819333081341234704, −5.26352441550839454361465841895, −5.24586828626391151538739432626, −5.21834574888762375507383875854, −4.58600274389060938481661853020, −4.48025035870240993949486311747, −4.14667961753109269745610518733, −4.13560308621852390306455175429, −3.63456794847818530194602873964, −3.42656927291329189050385780054, −3.30382241927306121630789328208, −3.23211690529046058739972414562, −2.68133330691268926924509747122, −2.53277124020744035810263799466, −2.51020122380301799864076599162, −2.36808440663608323630986088118, −1.80488282532410733670693829321, −1.64812963396263772664863426115, −1.29273365354815276872462150311, −1.21869551204738550102820898082, 0, 0, 0, 0, 1.21869551204738550102820898082, 1.29273365354815276872462150311, 1.64812963396263772664863426115, 1.80488282532410733670693829321, 2.36808440663608323630986088118, 2.51020122380301799864076599162, 2.53277124020744035810263799466, 2.68133330691268926924509747122, 3.23211690529046058739972414562, 3.30382241927306121630789328208, 3.42656927291329189050385780054, 3.63456794847818530194602873964, 4.13560308621852390306455175429, 4.14667961753109269745610518733, 4.48025035870240993949486311747, 4.58600274389060938481661853020, 5.21834574888762375507383875854, 5.24586828626391151538739432626, 5.26352441550839454361465841895, 5.36766461937819333081341234704, 5.85383996586136146947694339092, 5.86300646286894244226081144635, 6.03588414994209172792264608302, 6.03640927285934350426535462415, 6.36266633574154458096594057647

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