Properties

Degree 6
Conductor $ 5^{6} \cdot 47^{3} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 5·3-s + 3·4-s + 25·6-s + 45·7-s − 27·8-s − 49·9-s + 2·11-s + 15·12-s + 80·13-s + 225·14-s − 37·16-s + 39·17-s − 245·18-s − 24·19-s + 225·21-s + 10·22-s − 120·23-s − 135·24-s + 400·26-s − 353·27-s + 135·28-s − 184·29-s − 4·31-s + 53·32-s + 10·33-s + 195·34-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.962·3-s + 3/8·4-s + 1.70·6-s + 2.42·7-s − 1.19·8-s − 1.81·9-s + 0.0548·11-s + 0.360·12-s + 1.70·13-s + 4.29·14-s − 0.578·16-s + 0.556·17-s − 3.20·18-s − 0.289·19-s + 2.33·21-s + 0.0969·22-s − 1.08·23-s − 1.14·24-s + 3.01·26-s − 2.51·27-s + 0.911·28-s − 1.17·29-s − 0.0231·31-s + 0.292·32-s + 0.0527·33-s + 0.983·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(5^{6} \cdot 47^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{1175} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 5^{6} \cdot 47^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )$
$L(2)$  $\approx$  $18.97284347$
$L(\frac12)$  $\approx$  $18.97284347$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;47\}$, \(F_p\) is a polynomial of degree 6. If $p \in \{5,\;47\}$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad5 \( 1 \)
47$C_1$ \( ( 1 + p T )^{3} \)
good2$S_4\times C_2$ \( 1 - 5 T + 11 p T^{2} - 17 p^{2} T^{3} + 11 p^{4} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \)
3$S_4\times C_2$ \( 1 - 5 T + 74 T^{2} - 262 T^{3} + 74 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 - 45 T + 216 p T^{2} - 32022 T^{3} + 216 p^{4} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 2713 T^{2} + 13108 T^{3} + 2713 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 80 T + 7775 T^{2} - 355808 T^{3} + 7775 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 39 T + 10764 T^{2} - 269068 T^{3} + 10764 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 24 T + 19881 T^{2} + 312456 T^{3} + 19881 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 120 T + 28569 T^{2} + 1922592 T^{3} + 28569 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 184 T + 39951 T^{2} + 3957616 T^{3} + 39951 p^{3} T^{4} + 184 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 6029 T^{2} + 283720 p T^{3} + 6029 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 589 T + 7220 p T^{2} - 67144276 T^{3} + 7220 p^{4} T^{4} - 589 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 92 T + 4027 p T^{2} + 7995456 T^{3} + 4027 p^{4} T^{4} + 92 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 250 T + 171365 T^{2} - 42748948 T^{3} + 171365 p^{3} T^{4} - 250 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 459 T + 280872 T^{2} + 63986692 T^{3} + 280872 p^{3} T^{4} + 459 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 579 T + 228972 T^{2} - 94125566 T^{3} + 228972 p^{3} T^{4} - 579 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 267 T + 543684 T^{2} - 130418348 T^{3} + 543684 p^{3} T^{4} - 267 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 540 T + 119601 T^{2} + 142838224 T^{3} + 119601 p^{3} T^{4} - 540 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 749 T + 628750 T^{2} - 200987390 T^{3} + 628750 p^{3} T^{4} - 749 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1924 T + 2114435 T^{2} - 1560841600 T^{3} + 2114435 p^{3} T^{4} - 1924 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 805 T + 1187566 T^{2} - 671146982 T^{3} + 1187566 p^{3} T^{4} - 805 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 712 T + 1422097 T^{2} + 602850736 T^{3} + 1422097 p^{3} T^{4} + 712 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 835 T + 2039856 T^{2} - 1065240352 T^{3} + 2039856 p^{3} T^{4} - 835 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 2243 T + 3813620 T^{2} - 4231696160 T^{3} + 3813620 p^{3} T^{4} - 2243 p^{6} T^{5} + p^{9} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.215761791127584658506673134632, −8.008367927951789945534216600230, −7.994032970991170218435366373119, −7.81489064801472464435350272364, −7.33681484205522815136160844077, −6.70202954299820221412985015474, −6.43829939032793818162982488721, −6.10510903025516307938248923744, −5.81889215223919628430680848493, −5.67016518152890794734071159370, −5.21967868114184782331678362622, −5.11281542767261898123130230537, −4.94438638334739554779502450213, −4.34691846346306906734917614577, −4.15357313851768148657754762343, −4.02470683484685360097751601046, −3.49711640114080504664406331807, −3.29455529101673026242604248372, −3.17247929043010025150706433091, −2.30536584016938211582356448315, −2.20621294529197593402827895188, −1.94827046596921416600186491710, −1.32077201698659161111203176339, −0.789840490471609575049896819847, −0.47964979178292421347621110341, 0.47964979178292421347621110341, 0.789840490471609575049896819847, 1.32077201698659161111203176339, 1.94827046596921416600186491710, 2.20621294529197593402827895188, 2.30536584016938211582356448315, 3.17247929043010025150706433091, 3.29455529101673026242604248372, 3.49711640114080504664406331807, 4.02470683484685360097751601046, 4.15357313851768148657754762343, 4.34691846346306906734917614577, 4.94438638334739554779502450213, 5.11281542767261898123130230537, 5.21967868114184782331678362622, 5.67016518152890794734071159370, 5.81889215223919628430680848493, 6.10510903025516307938248923744, 6.43829939032793818162982488721, 6.70202954299820221412985015474, 7.33681484205522815136160844077, 7.81489064801472464435350272364, 7.994032970991170218435366373119, 8.008367927951789945534216600230, 8.215761791127584658506673134632

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