Properties

Label 6-3675e3-1.1-c1e3-0-1
Degree $6$
Conductor $49633171875$
Sign $-1$
Analytic cond. $25269.8$
Root an. cond. $5.41710$
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-s − 3·3-s + 3·6-s − 2·8-s + 6·9-s + 6·11-s − 6·13-s + 3·16-s − 6·18-s − 6·19-s − 6·22-s − 4·23-s + 6·24-s + 6·26-s − 10·27-s + 2·29-s − 2·31-s − 3·32-s − 18·33-s + 4·37-s + 6·38-s + 18·39-s − 2·41-s − 4·43-s + 4·46-s + 8·47-s − 9·48-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1.22·6-s − 0.707·8-s + 2·9-s + 1.80·11-s − 1.66·13-s + 3/4·16-s − 1.41·18-s − 1.37·19-s − 1.27·22-s − 0.834·23-s + 1.22·24-s + 1.17·26-s − 1.92·27-s + 0.371·29-s − 0.359·31-s − 0.530·32-s − 3.13·33-s + 0.657·37-s + 0.973·38-s + 2.88·39-s − 0.312·41-s − 0.609·43-s + 0.589·46-s + 1.16·47-s − 1.29·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 7^{6}\)
Sign: $-1$
Analytic conductor: \(25269.8\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3675} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
7 \( 1 \)
good2$S_4\times C_2$ \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 + 18 T + 311 T^{2} + 2732 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 255 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.81846677711049997898068960152, −7.62624284955739826504199252108, −7.46757460100901602501198410287, −7.21539296370568739724433809459, −6.67477566965535372968847849189, −6.50584057956923215638309814261, −6.47058772692979905940239932019, −6.45840492919719691405433746095, −5.99388072525571371806316475364, −5.67746706699558493605285790778, −5.46864384269813229538195131328, −5.09155416546173539219780430690, −5.07799165705330983961719586369, −4.51890590672999027272270943376, −4.29642790110606481260960268817, −4.21910684655347965701270558245, −3.88439785825654965994471494230, −3.58698898384042368185291722638, −3.04961765692660278457060654845, −2.84821970732624238087221305756, −2.39346676218986079088494122679, −2.04268395897040096406739916150, −1.65825899413591127426315052974, −1.17649483622796969392420270617, −1.13380744699443248209789838169, 0, 0, 0, 1.13380744699443248209789838169, 1.17649483622796969392420270617, 1.65825899413591127426315052974, 2.04268395897040096406739916150, 2.39346676218986079088494122679, 2.84821970732624238087221305756, 3.04961765692660278457060654845, 3.58698898384042368185291722638, 3.88439785825654965994471494230, 4.21910684655347965701270558245, 4.29642790110606481260960268817, 4.51890590672999027272270943376, 5.07799165705330983961719586369, 5.09155416546173539219780430690, 5.46864384269813229538195131328, 5.67746706699558493605285790778, 5.99388072525571371806316475364, 6.45840492919719691405433746095, 6.47058772692979905940239932019, 6.50584057956923215638309814261, 6.67477566965535372968847849189, 7.21539296370568739724433809459, 7.46757460100901602501198410287, 7.62624284955739826504199252108, 7.81846677711049997898068960152

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