Properties

Label 8-7800e4-1.1-c1e4-0-4
Degree $8$
Conductor $3.702\times 10^{15}$
Sign $1$
Analytic cond. $1.50482\times 10^{7}$
Root an. cond. $7.89197$
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  − 4·3-s − 7·7-s + 10·9-s + 11-s + 4·13-s − 3·17-s + 28·21-s − 5·23-s − 20·27-s + 8·29-s − 4·33-s − 5·37-s − 16·39-s + 7·41-s − 8·43-s − 10·47-s + 15·49-s + 12·51-s + 5·53-s + 2·59-s + 11·61-s − 70·63-s − 12·67-s + 20·69-s + 15·71-s + 10·73-s − 7·77-s + ⋯
L(s)  = 1  − 2.30·3-s − 2.64·7-s + 10/3·9-s + 0.301·11-s + 1.10·13-s − 0.727·17-s + 6.11·21-s − 1.04·23-s − 3.84·27-s + 1.48·29-s − 0.696·33-s − 0.821·37-s − 2.56·39-s + 1.09·41-s − 1.21·43-s − 1.45·47-s + 15/7·49-s + 1.68·51-s + 0.686·53-s + 0.260·59-s + 1.40·61-s − 8.81·63-s − 1.46·67-s + 2.40·69-s + 1.78·71-s + 1.17·73-s − 0.797·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.50482\times 10^{7}\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{4} \)
good7$C_2 \wr C_2\wr C_2$ \( 1 + p T + 34 T^{2} + 123 T^{3} + 362 T^{4} + 123 p T^{5} + 34 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - T + 24 T^{2} - 59 T^{3} + 282 T^{4} - 59 p T^{5} + 24 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 42 T^{2} + 117 T^{3} + 858 T^{4} + 117 p T^{5} + 42 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 72 T^{2} + 273 T^{3} + 2222 T^{4} + 273 p T^{5} + 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 96 T^{2} - 552 T^{3} + 4142 T^{4} - 552 p T^{5} + 96 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 94 T^{2} + 415 T^{3} + 4594 T^{4} + 415 p T^{5} + 94 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 156 T^{2} - 839 T^{3} + 9426 T^{4} - 839 p T^{5} + 156 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 152 T^{2} + 888 T^{3} + 9630 T^{4} + 888 p T^{5} + 152 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 206 T^{2} + 1358 T^{3} + 14818 T^{4} + 1358 p T^{5} + 206 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 158 T^{2} - 655 T^{3} + 11506 T^{4} - 655 p T^{5} + 158 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 150 T^{2} - 318 T^{3} + 11810 T^{4} - 318 p T^{5} + 150 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 222 T^{2} - 1885 T^{3} + 19674 T^{4} - 1885 p T^{5} + 222 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 244 T^{2} + 2188 T^{3} + 23462 T^{4} + 2188 p T^{5} + 244 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 254 T^{2} - 2665 T^{3} + 27242 T^{4} - 2665 p T^{5} + 254 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 280 T^{2} - 2022 T^{3} + 30254 T^{4} - 2022 p T^{5} + 280 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + T + 140 T^{2} + 185 T^{3} + 16966 T^{4} + 185 p T^{5} + 140 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 342 T^{2} + 3866 T^{3} + 40290 T^{4} + 3866 p T^{5} + 342 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 220 T^{2} - 2025 T^{3} + 23586 T^{4} - 2025 p T^{5} + 220 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + T + 116 T^{2} + 599 T^{3} + 10150 T^{4} + 599 p T^{5} + 116 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.07143934684204828004037915742, −5.63492599930712731556908024085, −5.53676987629127382346354593870, −5.49017157152100285261890666626, −5.33051518364156235967297098198, −4.82320837508619973334379590608, −4.74806920009578115721255664966, −4.67373987725238618088019300888, −4.65001056103789450696785205152, −4.07509907544541204728295473345, −3.90002217213662972162693252801, −3.89014396352738043094274084762, −3.86972592616098590059244680699, −3.32074589151223743370057936121, −3.24971373905285201124549963162, −3.11434351116710432370704701040, −3.08130469783594818068994250219, −2.41380220245548111843668225323, −2.16575678266499847351456010122, −2.16090022726238213684115631420, −2.14132555353948990598335533430, −1.22000292364139350708539200334, −1.18746793123677191815513396931, −1.14164525413407941619893673049, −0.982944537878363355669773880261, 0, 0, 0, 0, 0.982944537878363355669773880261, 1.14164525413407941619893673049, 1.18746793123677191815513396931, 1.22000292364139350708539200334, 2.14132555353948990598335533430, 2.16090022726238213684115631420, 2.16575678266499847351456010122, 2.41380220245548111843668225323, 3.08130469783594818068994250219, 3.11434351116710432370704701040, 3.24971373905285201124549963162, 3.32074589151223743370057936121, 3.86972592616098590059244680699, 3.89014396352738043094274084762, 3.90002217213662972162693252801, 4.07509907544541204728295473345, 4.65001056103789450696785205152, 4.67373987725238618088019300888, 4.74806920009578115721255664966, 4.82320837508619973334379590608, 5.33051518364156235967297098198, 5.49017157152100285261890666626, 5.53676987629127382346354593870, 5.63492599930712731556908024085, 6.07143934684204828004037915742

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