Properties

Label 8-912e4-1.1-c1e4-0-2
Degree $8$
Conductor $691798081536$
Sign $1$
Analytic cond. $2812.46$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 2·7-s + 3·9-s + 3·15-s − 3·17-s + 16·19-s + 2·21-s − 3·23-s − 2·25-s − 8·27-s − 5·29-s + 6·35-s − 7·41-s + 12·43-s − 9·45-s + 9·47-s − 9·49-s + 3·51-s + 17·53-s − 16·57-s + 9·59-s + 2·61-s − 6·63-s − 18·67-s + 3·69-s + 19·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.755·7-s + 9-s + 0.774·15-s − 0.727·17-s + 3.67·19-s + 0.436·21-s − 0.625·23-s − 2/5·25-s − 1.53·27-s − 0.928·29-s + 1.01·35-s − 1.09·41-s + 1.82·43-s − 1.34·45-s + 1.31·47-s − 9/7·49-s + 0.420·51-s + 2.33·53-s − 2.11·57-s + 1.17·59-s + 0.256·61-s − 0.755·63-s − 2.19·67-s + 0.361·69-s + 2.25·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2812.46\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6726047614\)
\(L(\frac12)\) \(\approx\) \(0.6726047614\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 + 15 T^{2} + 56 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 3 T + 35 T^{2} + 96 T^{3} + 786 T^{4} + 96 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 3 T - 19 T^{2} - 66 T^{3} + 24 T^{4} - 66 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 5 T - 31 T^{2} - 10 T^{3} + 1570 T^{4} - 10 p T^{5} - 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 45 T^{2} + 2024 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 39 T^{2} + 2120 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 7 T - 37 T^{2} + 28 T^{3} + 3706 T^{4} + 28 p T^{5} - 37 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 12 T + 55 T^{2} - 36 T^{3} - 120 T^{4} - 36 p T^{5} + 55 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 9 T + 125 T^{2} - 882 T^{3} + 8664 T^{4} - 882 p T^{5} + 125 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 17 T + 119 T^{2} - 1088 T^{3} + 10774 T^{4} - 1088 p T^{5} + 119 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 9 T - 49 T^{2} - 108 T^{3} + 8640 T^{4} - 108 p T^{5} - 49 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$D_4\times C_2$ \( 1 + 18 T + 225 T^{2} + 2106 T^{3} + 16436 T^{4} + 2106 p T^{5} + 225 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 19 T + 137 T^{2} - 1558 T^{3} + 19504 T^{4} - 1558 p T^{5} + 137 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^3$ \( 1 - 113 T^{2} + 7440 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 6 T - 3 T^{2} + 90 T^{3} - 5068 T^{4} + 90 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 9 T - 109 T^{2} - 108 T^{3} + 20970 T^{4} - 108 p T^{5} - 109 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 27 T + 473 T^{2} - 6210 T^{3} + 67062 T^{4} - 6210 p T^{5} + 473 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.39215106665300541929020180618, −7.18510710314316282867952367111, −6.67072296473410565744244921791, −6.64496481327307229168511369872, −6.50631149256716717840650498388, −5.97678808233427860600992962211, −5.79164439518417405738747662096, −5.70634139517173963621952683379, −5.35027901591189973234906641610, −5.12720929094183299820347678924, −5.10026514747757044155259599344, −4.60808663376182853960928729684, −4.32782704387718889544392684895, −3.97109712721241513517429721697, −3.86430467609551180482488048442, −3.76984374589751298527961054284, −3.34936401731513186087082592253, −3.29560626665196015554460041421, −2.78260059685038938493390978920, −2.34350109767437656404990814574, −2.23118072155728588046063924003, −1.68142902616280834193807810677, −1.11150631273840534052389915597, −0.966639648396388166099700232620, −0.25529920401353459718084040003, 0.25529920401353459718084040003, 0.966639648396388166099700232620, 1.11150631273840534052389915597, 1.68142902616280834193807810677, 2.23118072155728588046063924003, 2.34350109767437656404990814574, 2.78260059685038938493390978920, 3.29560626665196015554460041421, 3.34936401731513186087082592253, 3.76984374589751298527961054284, 3.86430467609551180482488048442, 3.97109712721241513517429721697, 4.32782704387718889544392684895, 4.60808663376182853960928729684, 5.10026514747757044155259599344, 5.12720929094183299820347678924, 5.35027901591189973234906641610, 5.70634139517173963621952683379, 5.79164439518417405738747662096, 5.97678808233427860600992962211, 6.50631149256716717840650498388, 6.64496481327307229168511369872, 6.67072296473410565744244921791, 7.18510710314316282867952367111, 7.39215106665300541929020180618

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