Properties

Label 6-4560e3-1.1-c1e3-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·5-s + 6·9-s + 2·11-s + 6·13-s + 9·15-s + 2·17-s − 3·19-s − 2·23-s + 6·25-s + 10·27-s + 4·29-s − 8·31-s + 6·33-s + 6·37-s + 18·39-s + 16·41-s + 4·43-s + 18·45-s + 14·47-s + 3·49-s + 6·51-s + 16·53-s + 6·55-s − 9·57-s − 4·59-s + 22·61-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 2·9-s + 0.603·11-s + 1.66·13-s + 2.32·15-s + 0.485·17-s − 0.688·19-s − 0.417·23-s + 6/5·25-s + 1.92·27-s + 0.742·29-s − 1.43·31-s + 1.04·33-s + 0.986·37-s + 2.88·39-s + 2.49·41-s + 0.609·43-s + 2.68·45-s + 2.04·47-s + 3/7·49-s + 0.840·51-s + 2.19·53-s + 0.809·55-s − 1.19·57-s − 0.520·59-s + 2.81·61-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

\(L(1)\) \(\approx\) \(22.64121523\)
\(L(\frac12)\) \(\approx\) \(22.64121523\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - 3 T^{2} + 4 T^{3} - 3 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 9 T^{2} - 8 T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 27 T^{2} - 120 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$D_{6}$ \( 1 - 2 T + 23 T^{2} - 60 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 2 T + 41 T^{2} + 84 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 4 T + 67 T^{2} - 220 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 8 T + 85 T^{2} + 448 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 99 T^{2} - 400 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 16 T + 183 T^{2} - 1348 T^{3} + 183 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} + 52 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 14 T + 177 T^{2} - 1292 T^{3} + 177 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 215 T^{2} - 1680 T^{3} + 215 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 81 T^{2} + 184 T^{3} + 81 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 22 T + 275 T^{2} - 2356 T^{3} + 275 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 + 16 T + 197 T^{2} + 1728 T^{3} + 197 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 14 T + 167 T^{2} - 1508 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 8 T - 19 T^{2} - 1168 T^{3} - 19 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 69 T^{2} - 388 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 4 T - 57 T^{2} + 1172 T^{3} - 57 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 6 T + 279 T^{2} + 1120 T^{3} + 279 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.49578735522363257772702704975, −7.31947812478263028776705568286, −6.85903045370354996312517214176, −6.70936327852825930391181139888, −6.26473274516624761747304014772, −6.20671825135323872467225093757, −5.85862737715754439959853550365, −5.77788073950248261964363644484, −5.50078450827291010944317078384, −5.23928256092379798539730292833, −4.70544754677320413233196100582, −4.46436373347290706958299627214, −4.22043161759661705658989496702, −3.96572390799856246632232345208, −3.69936337526964308894171453378, −3.63400611632924262343914540835, −3.15590086343900222633230438112, −2.68241167661707197994177492881, −2.68159067820545347368660584549, −2.22223622949683031548529545935, −1.93659036864651826985359685734, −1.92877499568284145264679222143, −1.13126660772651465550191689324, −0.901414376203660423871361901039, −0.850506354408407972590619575640, 0.850506354408407972590619575640, 0.901414376203660423871361901039, 1.13126660772651465550191689324, 1.92877499568284145264679222143, 1.93659036864651826985359685734, 2.22223622949683031548529545935, 2.68159067820545347368660584549, 2.68241167661707197994177492881, 3.15590086343900222633230438112, 3.63400611632924262343914540835, 3.69936337526964308894171453378, 3.96572390799856246632232345208, 4.22043161759661705658989496702, 4.46436373347290706958299627214, 4.70544754677320413233196100582, 5.23928256092379798539730292833, 5.50078450827291010944317078384, 5.77788073950248261964363644484, 5.85862737715754439959853550365, 6.20671825135323872467225093757, 6.26473274516624761747304014772, 6.70936327852825930391181139888, 6.85903045370354996312517214176, 7.31947812478263028776705568286, 7.49578735522363257772702704975

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