Properties

Label 6-7800e3-1.1-c1e3-0-2
Degree $6$
Conductor $474552000000$
Sign $1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s − 6·11-s − 3·13-s + 4·17-s − 2·19-s + 10·23-s − 10·27-s − 10·29-s + 12·31-s + 18·33-s − 6·37-s + 9·39-s − 10·41-s − 4·43-s + 16·47-s − 21·49-s − 12·51-s + 10·53-s + 6·57-s − 6·59-s + 2·61-s + 4·67-s − 30·69-s + 8·71-s + 6·73-s + 20·79-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 1.80·11-s − 0.832·13-s + 0.970·17-s − 0.458·19-s + 2.08·23-s − 1.92·27-s − 1.85·29-s + 2.15·31-s + 3.13·33-s − 0.986·37-s + 1.44·39-s − 1.56·41-s − 0.609·43-s + 2.33·47-s − 3·49-s − 1.68·51-s + 1.37·53-s + 0.794·57-s − 0.781·59-s + 0.256·61-s + 0.488·67-s − 3.61·69-s + 0.949·71-s + 0.702·73-s + 2.25·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.617729789\)
\(L(\frac12)\) \(\approx\) \(1.617729789\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$C_2$ \( ( 1 + p T^{2} )^{3} \)
11$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 112 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 31 T^{2} - 72 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 77 T^{2} - 380 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 43 T^{2} + 108 T^{3} + 43 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 12 T + 101 T^{2} - 584 T^{3} + 101 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 6 T + 83 T^{2} + 308 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 109 T^{2} + 280 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 16 T + 175 T^{2} - 1248 T^{3} + 175 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 115 T^{2} - 588 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$D_{6}$ \( 1 - 2 T + 151 T^{2} - 212 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 4 T + 73 T^{2} - 280 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 215 T^{2} - 1072 T^{3} + 215 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 71 T^{2} - 52 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 345 T^{2} - 3320 T^{3} + 345 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 187 T^{2} - 432 T^{3} + 187 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 22 T + 361 T^{2} + 3672 T^{3} + 361 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
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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

−6.96538411088603795364213743129, −6.57616106872550525270366432391, −6.52031669544587303673532198387, −6.44623029456296488930362747074, −5.68698586867143787130507874959, −5.68684936464556459182917286618, −5.63587959457817420763104512628, −5.20970765512806437204505101054, −5.10106057390892458554905689707, −5.05999070846689690846606003781, −4.61774557219760949959345048564, −4.43099735751405480050994468139, −4.30769739553919629288244953081, −3.63362864948348296696776105558, −3.62932430305048909538900389536, −3.27337446141092897764860245907, −2.93297134844054409058367291757, −2.69096946525478149690527944364, −2.47150415270460808491945962002, −1.93893175193214366695245804105, −1.63348048511831322267628921620, −1.58298707895077937324022508798, −0.67998279653304173848770218470, −0.67924522838979368550134217148, −0.37878694050747900260301198063, 0.37878694050747900260301198063, 0.67924522838979368550134217148, 0.67998279653304173848770218470, 1.58298707895077937324022508798, 1.63348048511831322267628921620, 1.93893175193214366695245804105, 2.47150415270460808491945962002, 2.69096946525478149690527944364, 2.93297134844054409058367291757, 3.27337446141092897764860245907, 3.62932430305048909538900389536, 3.63362864948348296696776105558, 4.30769739553919629288244953081, 4.43099735751405480050994468139, 4.61774557219760949959345048564, 5.05999070846689690846606003781, 5.10106057390892458554905689707, 5.20970765512806437204505101054, 5.63587959457817420763104512628, 5.68684936464556459182917286618, 5.68698586867143787130507874959, 6.44623029456296488930362747074, 6.52031669544587303673532198387, 6.57616106872550525270366432391, 6.96538411088603795364213743129

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