Properties

Label 8-1216e4-1.1-c1e4-0-19
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s − 7-s − 9-s − 7·11-s − 10·13-s + 2·15-s + 5·17-s − 4·19-s + 2·21-s − 8·23-s − 25-s + 2·27-s − 10·29-s − 6·31-s + 14·33-s + 35-s − 14·37-s + 20·39-s − 2·41-s − 3·43-s + 45-s − 3·47-s − 10·49-s − 10·51-s − 4·53-s + 7·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 0.377·7-s − 1/3·9-s − 2.11·11-s − 2.77·13-s + 0.516·15-s + 1.21·17-s − 0.917·19-s + 0.436·21-s − 1.66·23-s − 1/5·25-s + 0.384·27-s − 1.85·29-s − 1.07·31-s + 2.43·33-s + 0.169·35-s − 2.30·37-s + 3.20·39-s − 0.312·41-s − 0.457·43-s + 0.149·45-s − 0.437·47-s − 1.42·49-s − 1.40·51-s − 0.549·53-s + 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$D_{4}$ \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + T + 2 T^{2} - 9 T^{3} + 2 T^{4} - 9 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + T + 11 T^{2} + 40 T^{3} + 60 T^{4} + 40 p T^{5} + 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 4 p T^{2} + 203 T^{3} + 742 T^{4} + 203 p T^{5} + 4 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 59 T^{2} + 232 T^{3} + 852 T^{4} + 232 p T^{5} + 59 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + p T^{2} - 114 T^{3} + 698 T^{4} - 114 p T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 75 T^{2} + 420 T^{3} + 2456 T^{4} + 420 p T^{5} + 75 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 123 T^{2} + 712 T^{3} + 5108 T^{4} + 712 p T^{5} + 123 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 68 T^{2} + 142 T^{3} + 1782 T^{4} + 142 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 152 T^{2} + 1018 T^{3} + 7006 T^{4} + 1018 p T^{5} + 152 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 92 T^{2} + 54 T^{3} + 4694 T^{4} + 54 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 140 T^{2} + 291 T^{3} + 8278 T^{4} + 291 p T^{5} + 140 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 80 T^{2} + 247 T^{3} + 3038 T^{4} + 247 p T^{5} + 80 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 143 T^{2} + 252 T^{3} + 9032 T^{4} + 252 p T^{5} + 143 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 321 T^{2} + 3390 T^{3} + 30764 T^{4} + 3390 p T^{5} + 321 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 178 T^{2} - 521 T^{3} + 14618 T^{4} - 521 p T^{5} + 178 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 193 T^{2} + 1006 T^{3} + 16060 T^{4} + 1006 p T^{5} + 193 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 552 T^{2} + 6814 T^{3} + 65870 T^{4} + 6814 p T^{5} + 552 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 245 T^{2} - 1662 T^{3} + 25114 T^{4} - 1662 p T^{5} + 245 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 216 T^{2} - 1082 T^{3} + 18510 T^{4} - 1082 p T^{5} + 216 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 216 T^{2} + 212 T^{3} + 21054 T^{4} + 212 p T^{5} + 216 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 152 T^{2} + 224 T^{3} - 4082 T^{4} + 224 p T^{5} + 152 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 328 T^{2} + 1658 T^{3} + 45422 T^{4} + 1658 p T^{5} + 328 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.41738538466765150508131013236, −7.37336695932803392695267152499, −7.02582000669406280081917585506, −6.64577625281818495025655608618, −6.62034477251348148539150467029, −5.98844725780492197174645376794, −5.98259537352685995853648584553, −5.86104859391283632638325607903, −5.76443235836432001301413809826, −5.28543348576287487961505470997, −5.13931134350098002418834900576, −5.07548298225404164713768069460, −4.86288702390322254773005395949, −4.55161910337352866488735521570, −4.23802728891789322663845119229, −4.14076036916394777155130239983, −3.49136246005741852509466705711, −3.37583162159757549011012621570, −3.21257579409725015130037056997, −3.01150972928763480269215962994, −2.57715472574476540785951773797, −2.13520090082362835718224590119, −2.03845171612685715089886826272, −1.82685326359813927730044409360, −1.30931322109709111023474293258, 0, 0, 0, 0, 1.30931322109709111023474293258, 1.82685326359813927730044409360, 2.03845171612685715089886826272, 2.13520090082362835718224590119, 2.57715472574476540785951773797, 3.01150972928763480269215962994, 3.21257579409725015130037056997, 3.37583162159757549011012621570, 3.49136246005741852509466705711, 4.14076036916394777155130239983, 4.23802728891789322663845119229, 4.55161910337352866488735521570, 4.86288702390322254773005395949, 5.07548298225404164713768069460, 5.13931134350098002418834900576, 5.28543348576287487961505470997, 5.76443235836432001301413809826, 5.86104859391283632638325607903, 5.98259537352685995853648584553, 5.98844725780492197174645376794, 6.62034477251348148539150467029, 6.64577625281818495025655608618, 7.02582000669406280081917585506, 7.37336695932803392695267152499, 7.41738538466765150508131013236

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