Properties

Label 6-4008e3-1.1-c1e3-0-0
Degree $6$
Conductor $64384768512$
Sign $-1$
Analytic cond. $32780.4$
Root an. cond. $5.65721$
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  − 3·3-s + 6·5-s − 4·7-s + 6·9-s − 4·11-s + 6·13-s − 18·15-s + 4·19-s + 12·21-s − 24·23-s + 13·25-s − 10·27-s − 14·29-s − 4·31-s + 12·33-s − 24·35-s − 10·37-s − 18·39-s + 12·41-s − 8·43-s + 36·45-s + 3·49-s − 8·53-s − 24·55-s − 12·57-s + 4·59-s + 2·61-s + ⋯
L(s)  = 1  − 1.73·3-s + 2.68·5-s − 1.51·7-s + 2·9-s − 1.20·11-s + 1.66·13-s − 4.64·15-s + 0.917·19-s + 2.61·21-s − 5.00·23-s + 13/5·25-s − 1.92·27-s − 2.59·29-s − 0.718·31-s + 2.08·33-s − 4.05·35-s − 1.64·37-s − 2.88·39-s + 1.87·41-s − 1.21·43-s + 5.36·45-s + 3/7·49-s − 1.09·53-s − 3.23·55-s − 1.58·57-s + 0.520·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 167^{3}\)
Sign: $-1$
Analytic conductor: \(32780.4\)
Root analytic conductor: \(5.65721\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 167^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
167$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 62 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 4 T + 13 T^{2} + 40 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 29 T^{2} + 68 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 + 15 T^{2} + 54 T^{3} + 15 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 4 T + 41 T^{2} - 120 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 + 14 T + 115 T^{2} + 660 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 77 T^{2} + 216 T^{3} + 77 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 10 T + 131 T^{2} + 732 T^{3} + 131 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 12 T + 51 T^{2} - 66 T^{3} + 51 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 558 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 57 T^{2} - 268 T^{3} + 57 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 8 T + 115 T^{2} + 558 T^{3} + 115 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 145 T^{2} - 504 T^{3} + 145 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 51 T^{2} - 248 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 26 T + 413 T^{2} + 4038 T^{3} + 413 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 16 T + 149 T^{2} + 1232 T^{3} + 149 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 22 T + 319 T^{2} - 3316 T^{3} + 319 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 10 T + 137 T^{2} - 1302 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T - 7 T^{2} - 760 T^{3} - 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 6 T + 119 T^{2} + 148 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 159 T^{2} - 1316 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.969333816856581382141590735595, −7.41506980895942642208548855041, −7.22817135156938466297701531055, −7.18488490862490605272475568202, −6.40866250910239576938667840920, −6.37729859858106436269035302059, −6.35749189038882525130193159915, −6.01957869263193211444301971971, −5.82912065969384854962366515717, −5.81381608265759520226319229916, −5.39991663006317211410431754049, −5.37336160214590669487915737762, −5.22014281052842538230780686933, −4.47287349447737797604111409303, −4.40943342200377433208895373556, −3.90605862444746094724185916555, −3.69112474325232499769050563751, −3.51203875389296938405926345205, −3.27549923967169043655245302281, −2.55154450983491828023189593789, −2.26003828030938604311939245962, −2.15652732782009961347848103472, −1.61158611647973816581287197955, −1.58494208720156312370006693725, −1.22290331148981767870813992561, 0, 0, 0, 1.22290331148981767870813992561, 1.58494208720156312370006693725, 1.61158611647973816581287197955, 2.15652732782009961347848103472, 2.26003828030938604311939245962, 2.55154450983491828023189593789, 3.27549923967169043655245302281, 3.51203875389296938405926345205, 3.69112474325232499769050563751, 3.90605862444746094724185916555, 4.40943342200377433208895373556, 4.47287349447737797604111409303, 5.22014281052842538230780686933, 5.37336160214590669487915737762, 5.39991663006317211410431754049, 5.81381608265759520226319229916, 5.82912065969384854962366515717, 6.01957869263193211444301971971, 6.35749189038882525130193159915, 6.37729859858106436269035302059, 6.40866250910239576938667840920, 7.18488490862490605272475568202, 7.22817135156938466297701531055, 7.41506980895942642208548855041, 7.969333816856581382141590735595

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