Properties

Label 6-3549e3-1.1-c1e3-0-5
Degree $6$
Conductor $44701078149$
Sign $-1$
Analytic cond. $22758.7$
Root an. cond. $5.32343$
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  − 2·2-s − 3·3-s + 4-s + 6·6-s − 3·7-s + 8-s + 6·9-s − 8·11-s − 3·12-s + 6·14-s − 5·16-s + 4·17-s − 12·18-s − 7·19-s + 9·21-s + 16·22-s + 9·23-s − 3·24-s − 2·25-s − 10·27-s − 3·28-s − 7·29-s − 7·31-s + 8·32-s + 24·33-s − 8·34-s + 6·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s + 2.44·6-s − 1.13·7-s + 0.353·8-s + 2·9-s − 2.41·11-s − 0.866·12-s + 1.60·14-s − 5/4·16-s + 0.970·17-s − 2.82·18-s − 1.60·19-s + 1.96·21-s + 3.41·22-s + 1.87·23-s − 0.612·24-s − 2/5·25-s − 1.92·27-s − 0.566·28-s − 1.29·29-s − 1.25·31-s + 1.41·32-s + 4.17·33-s − 1.37·34-s + 36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 7^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(22758.7\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3549} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good2$A_4\times C_2$ \( 1 + p T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 2 T^{2} + 13 T^{3} + 2 p T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 8 T + 50 T^{2} + 181 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 4 T + 52 T^{2} - 135 T^{3} + 52 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 7 T + 56 T^{2} + 219 T^{3} + 56 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 9 T + 83 T^{2} - 415 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 7 T + 73 T^{2} + 411 T^{3} + 73 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 7 T + 53 T^{2} + 153 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 98 T^{2} - 13 T^{3} + 98 p T^{4} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 2 T + 55 T^{2} + 36 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 19 T + 245 T^{2} + 1863 T^{3} + 245 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 17 T + 155 T^{2} + 1051 T^{3} + 155 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 13 T + 198 T^{2} - 1365 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 3 T + 167 T^{2} + 329 T^{3} + 167 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 13 T + 222 T^{2} - 1573 T^{3} + 222 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 5 T + 127 T^{2} - 275 T^{3} + 127 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 8 T + 100 T^{2} - 621 T^{3} + 100 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 2 T + 138 T^{2} + 5 p T^{3} + 138 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + T - 53 T^{2} - 179 T^{3} - 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 2 T + 64 T^{2} - 561 T^{3} + 64 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 19 T + 266 T^{2} + 2363 T^{3} + 266 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 27 T + 521 T^{2} - 5837 T^{3} + 521 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.980468891201453987807055280445, −7.85595320059179725503021494204, −7.46630934932360370776404042176, −6.98749777033367017660818717495, −6.95607137472504086942801920303, −6.91725220837475941356058933770, −6.47939429052555343443938303947, −6.44023161752477443162316741878, −5.86848707184440365128728627494, −5.60334749735965075560252124079, −5.35214894765144139250994531508, −5.32233869505146113694213276181, −5.18078879247986688435413125853, −4.65008980045169729425752255918, −4.40327376802207253781893107774, −4.28694486483565235698654136362, −3.51544789660644532044603906601, −3.46678786206920536405287757848, −3.33044677494299758005205940732, −2.73782827203641924204809649823, −2.25329671588994649368203761105, −2.17863912443221202409894227067, −1.73049021298553151312014415776, −1.17560675488801290248571201276, −0.76882570216712483730003469916, 0, 0, 0, 0.76882570216712483730003469916, 1.17560675488801290248571201276, 1.73049021298553151312014415776, 2.17863912443221202409894227067, 2.25329671588994649368203761105, 2.73782827203641924204809649823, 3.33044677494299758005205940732, 3.46678786206920536405287757848, 3.51544789660644532044603906601, 4.28694486483565235698654136362, 4.40327376802207253781893107774, 4.65008980045169729425752255918, 5.18078879247986688435413125853, 5.32233869505146113694213276181, 5.35214894765144139250994531508, 5.60334749735965075560252124079, 5.86848707184440365128728627494, 6.44023161752477443162316741878, 6.47939429052555343443938303947, 6.91725220837475941356058933770, 6.95607137472504086942801920303, 6.98749777033367017660818717495, 7.46630934932360370776404042176, 7.85595320059179725503021494204, 7.980468891201453987807055280445

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