Properties

Label 6-2320e3-1.1-c1e3-0-8
Degree $6$
Conductor $12487168000$
Sign $-1$
Analytic cond. $6357.63$
Root an. cond. $4.30410$
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  − 2·3-s + 3·5-s − 4·7-s − 9-s − 2·11-s − 2·13-s − 6·15-s − 4·17-s + 10·19-s + 8·21-s − 16·23-s + 6·25-s + 4·27-s − 3·29-s + 14·31-s + 4·33-s − 12·35-s − 8·37-s + 4·39-s − 2·41-s − 2·43-s − 3·45-s − 14·47-s − 5·49-s + 8·51-s + 6·53-s − 6·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.34·5-s − 1.51·7-s − 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.54·15-s − 0.970·17-s + 2.29·19-s + 1.74·21-s − 3.33·23-s + 6/5·25-s + 0.769·27-s − 0.557·29-s + 2.51·31-s + 0.696·33-s − 2.02·35-s − 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.304·43-s − 0.447·45-s − 2.04·47-s − 5/7·49-s + 1.12·51-s + 0.824·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 29^{3}\)
Sign: $-1$
Analytic conductor: \(6357.63\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
29$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 4 T + 3 p T^{2} + 52 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 25 T^{2} + 48 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 10 T + 85 T^{2} - 400 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 16 T + 145 T^{2} + 36 p T^{3} + 145 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 153 T^{2} - 944 T^{3} + 153 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 T + 87 T^{2} + 500 T^{3} + 87 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 39 T^{2} + 396 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T - 3 T^{2} + 176 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 14 T + 201 T^{2} + 1392 T^{3} + 201 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 155 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 75 T^{2} + 516 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 28 T + 453 T^{2} + 4468 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 28 T + 389 T^{2} + 3704 T^{3} + 389 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 16 T + 119 T^{2} + 636 T^{3} + 119 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 149 T^{2} - 488 T^{3} + 149 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 1844 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 10 T + 279 T^{2} + 1740 T^{3} + 279 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 223 T^{2} - 1628 T^{3} + 223 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.479608262859881067241113218834, −7.990946730738580309494609549983, −7.83099811292399877432628865849, −7.66507324989410495835813437719, −7.17607561982663976030219482907, −6.95933501230260744099342894564, −6.74566487931614398194963253647, −6.29804611003827923551208339098, −6.20615141065405095154781385533, −6.10145999244611337077111590056, −5.62802496321630759575767822679, −5.58656973697870970192704263805, −5.50618569288743473641584916961, −4.93110585012273394135103594309, −4.57508898086561690810429420236, −4.46502584156946472834945228109, −4.22950582484733885682477022627, −3.42120148486559331899263347288, −3.34236184894800141692173368154, −2.89992172422406699571306929457, −2.87271802482425086336827568496, −2.45410787754701637042230364294, −1.87917261777697366304654108321, −1.47880296336066571406827809111, −1.34756792108083830012320212142, 0, 0, 0, 1.34756792108083830012320212142, 1.47880296336066571406827809111, 1.87917261777697366304654108321, 2.45410787754701637042230364294, 2.87271802482425086336827568496, 2.89992172422406699571306929457, 3.34236184894800141692173368154, 3.42120148486559331899263347288, 4.22950582484733885682477022627, 4.46502584156946472834945228109, 4.57508898086561690810429420236, 4.93110585012273394135103594309, 5.50618569288743473641584916961, 5.58656973697870970192704263805, 5.62802496321630759575767822679, 6.10145999244611337077111590056, 6.20615141065405095154781385533, 6.29804611003827923551208339098, 6.74566487931614398194963253647, 6.95933501230260744099342894564, 7.17607561982663976030219482907, 7.66507324989410495835813437719, 7.83099811292399877432628865849, 7.990946730738580309494609549983, 8.479608262859881067241113218834

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