Properties

Label 6-825e3-1.1-c1e3-0-6
Degree $6$
Conductor $561515625$
Sign $-1$
Analytic cond. $285.886$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 4·4-s − 9·6-s − 8·7-s − 4·8-s + 6·9-s − 3·11-s + 12·12-s − 6·13-s + 24·14-s + 3·16-s − 4·17-s − 18·18-s − 2·19-s − 24·21-s + 9·22-s − 12·23-s − 12·24-s + 18·26-s + 10·27-s − 32·28-s − 8·29-s + 8·31-s + 32-s − 9·33-s + 12·34-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 2·4-s − 3.67·6-s − 3.02·7-s − 1.41·8-s + 2·9-s − 0.904·11-s + 3.46·12-s − 1.66·13-s + 6.41·14-s + 3/4·16-s − 0.970·17-s − 4.24·18-s − 0.458·19-s − 5.23·21-s + 1.91·22-s − 2.50·23-s − 2.44·24-s + 3.53·26-s + 1.92·27-s − 6.04·28-s − 1.48·29-s + 1.43·31-s + 0.176·32-s − 1.56·33-s + 2.05·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(285.886\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 8 T + 37 T^{2} + 116 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 6 T + 11 T^{2} + 8 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 23 T^{2} + 20 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 432 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 624 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 145 T^{2} + 692 T^{3} + 145 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 127 T^{2} - 576 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T + 185 T^{2} + 1288 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 18 T + 279 T^{2} + 2536 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 2 T + 245 T^{2} - 328 T^{3} + 245 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 255 T^{2} - 348 T^{3} + 255 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 259 T^{2} - 1424 T^{3} + 259 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.642618645744077532626415376066, −9.153010968813751776226261488951, −9.060005868768446311906458682110, −8.827688934420347286038269668205, −8.492362227673343736911577944408, −8.162307526514416849187154306414, −8.052947122976830871173468382930, −7.58641999983254212492816478410, −7.48239538220345664434792632423, −7.00717280563667899036143555789, −6.76909145900880626992601319300, −6.58698742743786968829934548755, −6.21760680023177783022639434745, −6.03644935278392690009694102950, −5.30010138021793595739629337946, −5.20577822701259887684677418348, −4.39573475830108186797434377355, −4.15681673162548209544048057646, −3.90239618656781203444884810085, −3.32735084440142798165674678773, −2.93043922315751549885649373895, −2.77787129063554322030528680232, −2.56383985908226904712142055731, −1.80804594194448207286975864668, −1.70933572110942068536528085561, 0, 0, 0, 1.70933572110942068536528085561, 1.80804594194448207286975864668, 2.56383985908226904712142055731, 2.77787129063554322030528680232, 2.93043922315751549885649373895, 3.32735084440142798165674678773, 3.90239618656781203444884810085, 4.15681673162548209544048057646, 4.39573475830108186797434377355, 5.20577822701259887684677418348, 5.30010138021793595739629337946, 6.03644935278392690009694102950, 6.21760680023177783022639434745, 6.58698742743786968829934548755, 6.76909145900880626992601319300, 7.00717280563667899036143555789, 7.48239538220345664434792632423, 7.58641999983254212492816478410, 8.052947122976830871173468382930, 8.162307526514416849187154306414, 8.492362227673343736911577944408, 8.827688934420347286038269668205, 9.060005868768446311906458682110, 9.153010968813751776226261488951, 9.642618645744077532626415376066

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