Properties

Label 6-5520e3-1.1-c1e3-0-0
Degree $6$
Conductor $168196608000$
Sign $1$
Analytic cond. $85634.4$
Root an. cond. $6.63908$
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 − 3·5-s − 2·7-s + 6·9-s − 4·11-s + 2·13-s + 9·15-s − 10·19-s + 6·21-s − 3·23-s + 6·25-s − 10·27-s − 10·31-s + 12·33-s + 6·35-s + 2·37-s − 6·39-s + 8·41-s − 16·43-s − 18·45-s + 4·47-s − 2·49-s + 8·53-s + 12·55-s + 30·57-s + 10·61-s − 12·63-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 1.20·11-s + 0.554·13-s + 2.32·15-s − 2.29·19-s + 1.30·21-s − 0.625·23-s + 6/5·25-s − 1.92·27-s − 1.79·31-s + 2.08·33-s + 1.01·35-s + 0.328·37-s − 0.960·39-s + 1.24·41-s − 2.43·43-s − 2.68·45-s + 0.583·47-s − 2/7·49-s + 1.09·53-s + 1.61·55-s + 3.97·57-s + 1.28·61-s − 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(85634.4\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2358946503\)
\(L(\frac12)\) \(\approx\) \(0.2358946503\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 + 2 T + 6 T^{2} + 4 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 19 T^{2} + 40 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 21 T^{2} - 64 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 30 T^{2} + 18 T^{3} + 30 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 10 T + 71 T^{2} + 328 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 66 T^{2} - 18 T^{3} + 66 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 10 T + 110 T^{2} + 616 T^{3} + 110 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 48 T^{2} - 256 T^{3} + 48 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 8 T + 22 T^{2} - 74 T^{3} + 22 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 16 T + 149 T^{2} + 1072 T^{3} + 149 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 91 T^{2} - 160 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 58 T^{2} - 266 T^{3} + 58 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 156 T^{2} - 18 T^{3} + 156 p T^{4} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 10 T + 161 T^{2} - 928 T^{3} + 161 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 10 T + 218 T^{2} + 1336 T^{3} + 218 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 112 T^{2} - 554 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$D_{6}$ \( 1 - 18 T + 153 T^{2} - 1136 T^{3} + 153 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 14 T + 3 p T^{2} + 2116 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 232 T^{2} + 1432 T^{3} + 232 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 67 T^{2} + 556 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 20 T + 203 T^{2} + 1544 T^{3} + 203 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.23447779617899550283364496424, −6.93146970404755496431654384597, −6.62398934185607041365153503565, −6.44200395169925744241350885666, −6.42757920877965860724614371320, −6.01254943833673489252234234595, −5.64636316829900980952359678357, −5.49001830752502589657451499027, −5.39615097898362078985309661355, −5.16592220236755315666087666599, −4.60486045149467756356698393001, −4.49411956088056634368087831401, −4.29056757281591785483366264030, −4.02008612687333361725807941306, −3.77661811621993423556503932306, −3.61996602786703754091005141598, −3.07016059682869520306203526823, −2.94862976774238059131393036388, −2.56927071739488823782904089074, −1.96437124743490682258231862530, −1.91550436238486790551727118414, −1.57806176310532293795507537439, −0.76080912806436791773259687589, −0.56402317618019820284477534291, −0.18496868059260596846442361952, 0.18496868059260596846442361952, 0.56402317618019820284477534291, 0.76080912806436791773259687589, 1.57806176310532293795507537439, 1.91550436238486790551727118414, 1.96437124743490682258231862530, 2.56927071739488823782904089074, 2.94862976774238059131393036388, 3.07016059682869520306203526823, 3.61996602786703754091005141598, 3.77661811621993423556503932306, 4.02008612687333361725807941306, 4.29056757281591785483366264030, 4.49411956088056634368087831401, 4.60486045149467756356698393001, 5.16592220236755315666087666599, 5.39615097898362078985309661355, 5.49001830752502589657451499027, 5.64636316829900980952359678357, 6.01254943833673489252234234595, 6.42757920877965860724614371320, 6.44200395169925744241350885666, 6.62398934185607041365153503565, 6.93146970404755496431654384597, 7.23447779617899550283364496424

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