Properties

Label 6-7800e3-1.1-c1e3-0-11
Degree $6$
Conductor $474552000000$
Sign $-1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 7-s + 6·9-s − 3·11-s + 3·13-s − 5·17-s − 2·19-s − 3·21-s − 4·23-s + 10·27-s − 29-s − 11·31-s − 9·33-s − 10·37-s + 9·39-s − 16·41-s + 11·47-s − 5·49-s − 15·51-s − 19·53-s − 6·57-s − 3·59-s − 21·61-s − 6·63-s − 67-s − 12·69-s + 2·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.377·7-s + 2·9-s − 0.904·11-s + 0.832·13-s − 1.21·17-s − 0.458·19-s − 0.654·21-s − 0.834·23-s + 1.92·27-s − 0.185·29-s − 1.97·31-s − 1.56·33-s − 1.64·37-s + 1.44·39-s − 2.49·41-s + 1.60·47-s − 5/7·49-s − 2.10·51-s − 2.60·53-s − 0.794·57-s − 0.390·59-s − 2.68·61-s − 0.755·63-s − 0.122·67-s − 1.44·69-s + 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(241610.\)
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: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + T + 6 T^{2} - 11 T^{3} + 6 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 32 T^{2} + 65 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 5 T + 22 T^{2} + 33 T^{3} + 22 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 53 T^{2} + 72 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T - 3 T^{2} - 84 T^{3} - 3 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T + 74 T^{2} + 53 T^{3} + 74 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 11 T + 120 T^{2} + 701 T^{3} + 120 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 10 T + 55 T^{2} + 184 T^{3} + 55 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 16 T + 175 T^{2} + 1212 T^{3} + 175 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 45 T^{2} - 268 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 11 T + 166 T^{2} - 1047 T^{3} + 166 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 19 T + 274 T^{2} + 2235 T^{3} + 274 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 3 T - 30 T^{2} - 321 T^{3} - 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 21 T + 314 T^{2} + 2777 T^{3} + 314 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + T + 164 T^{2} + 171 T^{3} + 164 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 2 T + 65 T^{2} + 512 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 2 T + 63 T^{2} - 568 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 185 T^{2} + 1292 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 130 T^{2} - 303 T^{3} + 130 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 16 T + 107 T^{2} + 752 T^{3} + 107 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 10 T + 263 T^{2} - 1932 T^{3} + 263 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.46752314459156243723744748820, −7.01473243499615853878854932897, −6.91221513134643170505908712853, −6.71454363747251563286837585232, −6.42466918967995083272836673623, −6.10339624602389522349299584395, −6.04931469524107199889656781473, −5.60803903917434204237023926657, −5.35189552246265165594389930967, −5.27065723274372976964573113837, −4.66445312053684309837626136192, −4.60141199254077946656637275057, −4.53389517772733922673218354643, −4.08931592146035194875149623655, −3.68696826474404006952361730615, −3.66861186283513515364350152601, −3.31772121035556693787963817566, −3.14250107078741104455590763966, −2.99656402603008311346640098488, −2.44015428937321343220332589783, −2.24494302755815531601100852812, −2.08616424119685487402809240560, −1.62830310060678144196861963437, −1.40478154151444695496395476149, −1.29964482889531259012845619998, 0, 0, 0, 1.29964482889531259012845619998, 1.40478154151444695496395476149, 1.62830310060678144196861963437, 2.08616424119685487402809240560, 2.24494302755815531601100852812, 2.44015428937321343220332589783, 2.99656402603008311346640098488, 3.14250107078741104455590763966, 3.31772121035556693787963817566, 3.66861186283513515364350152601, 3.68696826474404006952361730615, 4.08931592146035194875149623655, 4.53389517772733922673218354643, 4.60141199254077946656637275057, 4.66445312053684309837626136192, 5.27065723274372976964573113837, 5.35189552246265165594389930967, 5.60803903917434204237023926657, 6.04931469524107199889656781473, 6.10339624602389522349299584395, 6.42466918967995083272836673623, 6.71454363747251563286837585232, 6.91221513134643170505908712853, 7.01473243499615853878854932897, 7.46752314459156243723744748820

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