Properties

Label 6-4560e3-1.1-c1e3-0-1
Degree $6$
Conductor $94818816000$
Sign $1$
Analytic cond. $48275.3$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s + 6·9-s − 4·11-s + 6·13-s + 9·15-s + 2·17-s + 3·19-s − 4·23-s + 6·25-s − 10·27-s + 2·29-s − 8·31-s + 12·33-s + 6·37-s − 18·39-s − 2·41-s − 12·43-s − 18·45-s + 4·47-s − 7·49-s − 6·51-s + 2·53-s + 12·55-s − 9·57-s − 12·59-s + 14·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s + 2·9-s − 1.20·11-s + 1.66·13-s + 2.32·15-s + 0.485·17-s + 0.688·19-s − 0.834·23-s + 6/5·25-s − 1.92·27-s + 0.371·29-s − 1.43·31-s + 2.08·33-s + 0.986·37-s − 2.88·39-s − 0.312·41-s − 1.82·43-s − 2.68·45-s + 0.583·47-s − 49-s − 0.840·51-s + 0.274·53-s + 1.61·55-s − 1.19·57-s − 1.56·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(48275.3\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.440756965\)
\(L(\frac12)\) \(\approx\) \(1.440756965\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + p T^{2} - 16 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 7 T^{2} - 8 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 6 T + 37 T^{2} - 120 T^{3} + 37 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 27 T^{2} - 52 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 41 T^{2} + 40 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 8 T + 89 T^{2} + 448 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 109 T^{2} - 408 T^{3} + 109 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 12 T + 163 T^{2} + 1024 T^{3} + 163 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 121 T^{2} - 344 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 2 T + 135 T^{2} - 196 T^{3} + 135 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 169 T^{2} + 1128 T^{3} + 169 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 151 T^{2} - 1132 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 4 T + 105 T^{2} + 408 T^{3} + 105 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 8 T + 109 T^{2} + 368 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{3} \)
79$S_4\times C_2$ \( 1 - 4 T + 141 T^{2} - 504 T^{3} + 141 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T - 35 T^{2} - 1464 T^{3} - 35 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 141 T^{2} - 152 T^{3} + 141 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 30 T + 577 T^{2} - 6664 T^{3} + 577 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.47641027159507669913966469524, −7.14895817382775261480952973757, −6.76375131074506409994416460273, −6.69937079791830339775835528432, −6.24567174371663540764457455098, −6.16938542931036424871859811240, −5.95015214540643568874222890407, −5.56715452265414334711730264005, −5.45471184241227539161713615691, −5.12633724426551357687722013577, −4.90857428746292522172906084883, −4.60893112676358788196324467499, −4.53121116695540434977518117393, −3.99384385244426311562481846001, −3.80262046352314532635788470346, −3.59554687344549631003018020592, −3.31549814423213006283430285465, −3.03650784455289186903572829244, −2.75854425724953520010649723259, −1.91836667308977616396531829929, −1.90334398580523753014845680352, −1.61760611301596332199263100631, −0.801575136536248287435852033366, −0.66301121612278665301158155634, −0.44156962705337997371582218644, 0.44156962705337997371582218644, 0.66301121612278665301158155634, 0.801575136536248287435852033366, 1.61760611301596332199263100631, 1.90334398580523753014845680352, 1.91836667308977616396531829929, 2.75854425724953520010649723259, 3.03650784455289186903572829244, 3.31549814423213006283430285465, 3.59554687344549631003018020592, 3.80262046352314532635788470346, 3.99384385244426311562481846001, 4.53121116695540434977518117393, 4.60893112676358788196324467499, 4.90857428746292522172906084883, 5.12633724426551357687722013577, 5.45471184241227539161713615691, 5.56715452265414334711730264005, 5.95015214540643568874222890407, 6.16938542931036424871859811240, 6.24567174371663540764457455098, 6.69937079791830339775835528432, 6.76375131074506409994416460273, 7.14895817382775261480952973757, 7.47641027159507669913966469524

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