Properties

Label 6-1840e3-1.1-c1e3-0-2
Degree $6$
Conductor $6229504000$
Sign $-1$
Analytic cond. $3171.64$
Root an. cond. $3.83307$
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-s − 3·5-s − 3·7-s + 9-s − 3·11-s − 13-s + 3·15-s − 7·17-s − 3·19-s + 3·21-s + 3·23-s + 6·25-s + 5·27-s − 4·29-s + 5·31-s + 3·33-s + 9·35-s − 2·37-s + 39-s + 41-s − 24·43-s − 3·45-s + 14·47-s + 9·49-s + 7·51-s − 18·53-s + 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.774·15-s − 1.69·17-s − 0.688·19-s + 0.654·21-s + 0.625·23-s + 6/5·25-s + 0.962·27-s − 0.742·29-s + 0.898·31-s + 0.522·33-s + 1.52·35-s − 0.328·37-s + 0.160·39-s + 0.156·41-s − 3.65·43-s − 0.447·45-s + 2.04·47-s + 9/7·49-s + 0.980·51-s − 2.47·53-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(3171.64\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T - 2 p T^{3} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 3 T - 22 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T - 6 T^{2} - 78 T^{3} - 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + T + 24 T^{2} + 8 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 7 T + 58 T^{2} + 220 T^{3} + 58 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 36 T^{2} + 50 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 55 T^{2} + 256 T^{3} + 55 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 86 T^{2} - 302 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 2 T + 71 T^{2} + 116 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - T + 64 T^{2} + 104 T^{3} + 64 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
47$S_4\times C_2$ \( 1 - 14 T + 145 T^{2} - 1028 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 + 14 T + 205 T^{2} + 1508 T^{3} + 205 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 26 T^{2} + 404 T^{3} + 26 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$D_{6}$ \( 1 + 8 T + 57 T^{2} + 688 T^{3} + 57 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 11 T + 244 T^{2} + 1586 T^{3} + 244 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 8 T + 179 T^{2} + 920 T^{3} + 179 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 4 T - 3 T^{2} + 520 T^{3} - 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 229 T^{2} + 1232 T^{3} + 229 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 219 T^{2} - 2052 T^{3} + 219 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 33 T + 570 T^{2} + 6568 T^{3} + 570 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.671000380389074055058138627513, −8.267085106070241352926660708959, −8.067792522882624271632043906777, −7.902244953715744039878273262532, −7.41031869553070581474697889156, −7.34372772175870721098056832710, −6.87380197800133543890469895292, −6.72020924763452087665181835592, −6.57360718025229415866192774644, −6.39155683924310895797136880991, −5.94356142229100703688422690129, −5.63101603209270277629491108185, −5.22322293436153291608865640477, −5.07185923159146725553510670007, −4.65560005186299229420278065265, −4.38710566350799376266440385759, −4.31951891229441644583173274331, −3.85681347769988548252020946868, −3.53261079916272994701441132978, −3.04713768463449540823716191034, −2.86783702482405152928881422469, −2.66822144317941243329337021161, −2.18143918793632073363691886717, −1.35504653420528706663932098210, −1.35246250508633082812693008925, 0, 0, 0, 1.35246250508633082812693008925, 1.35504653420528706663932098210, 2.18143918793632073363691886717, 2.66822144317941243329337021161, 2.86783702482405152928881422469, 3.04713768463449540823716191034, 3.53261079916272994701441132978, 3.85681347769988548252020946868, 4.31951891229441644583173274331, 4.38710566350799376266440385759, 4.65560005186299229420278065265, 5.07185923159146725553510670007, 5.22322293436153291608865640477, 5.63101603209270277629491108185, 5.94356142229100703688422690129, 6.39155683924310895797136880991, 6.57360718025229415866192774644, 6.72020924763452087665181835592, 6.87380197800133543890469895292, 7.34372772175870721098056832710, 7.41031869553070581474697889156, 7.902244953715744039878273262532, 8.067792522882624271632043906777, 8.267085106070241352926660708959, 8.671000380389074055058138627513

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