Properties

Label 6-7800e3-1.1-c1e3-0-8
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 7-s + 6·9-s − 3·11-s − 3·13-s + 5·17-s − 2·19-s − 3·21-s + 4·23-s − 10·27-s − 29-s − 11·31-s + 9·33-s + 10·37-s + 9·39-s − 16·41-s − 11·47-s − 5·49-s − 15·51-s + 19·53-s + 6·57-s − 3·59-s − 21·61-s + 6·63-s + 67-s − 12·69-s + 2·71-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.377·7-s + 2·9-s − 0.904·11-s − 0.832·13-s + 1.21·17-s − 0.458·19-s − 0.654·21-s + 0.834·23-s − 1.92·27-s − 0.185·29-s − 1.97·31-s + 1.56·33-s + 1.64·37-s + 1.44·39-s − 2.49·41-s − 1.60·47-s − 5/7·49-s − 2.10·51-s + 2.60·53-s + 0.794·57-s − 0.390·59-s − 2.68·61-s + 0.755·63-s + 0.122·67-s − 1.44·69-s + 0.237·71-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: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - T + 6 T^{2} + 11 T^{3} + 6 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 32 T^{2} + 65 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 5 T + 22 T^{2} - 33 T^{3} + 22 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 53 T^{2} + 72 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T - 3 T^{2} + 84 T^{3} - 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T + 74 T^{2} + 53 T^{3} + 74 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 11 T + 120 T^{2} + 701 T^{3} + 120 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 55 T^{2} - 184 T^{3} + 55 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 16 T + 175 T^{2} + 1212 T^{3} + 175 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 45 T^{2} + 268 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 11 T + 166 T^{2} + 1047 T^{3} + 166 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 19 T + 274 T^{2} - 2235 T^{3} + 274 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 3 T - 30 T^{2} - 321 T^{3} - 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 21 T + 314 T^{2} + 2777 T^{3} + 314 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - T + 164 T^{2} - 171 T^{3} + 164 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 2 T + 65 T^{2} + 512 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 2 T + 63 T^{2} + 568 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 185 T^{2} + 1292 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 7 T + 130 T^{2} + 303 T^{3} + 130 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 16 T + 107 T^{2} + 752 T^{3} + 107 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 10 T + 263 T^{2} + 1932 T^{3} + 263 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.30123541572636076832801252673, −6.95687309128270076397110134511, −6.88427270806825451833729780774, −6.67446114524337539152497094972, −6.17149092093856526155808040517, −6.14357637020982794866056435722, −5.84811302662305641369194610051, −5.51136156742285234112041417663, −5.50429815028034712654363405230, −5.24243689373481727927031955730, −4.82192485556139645333758822505, −4.81565279272108814951564343679, −4.76887402181781235003506978590, −4.10395983870123998653375327020, −4.06546464417264798662282347153, −3.84233490312395565961706473238, −3.17818212786065565609748409028, −3.10859223609433760355684432072, −3.09696383709189731672943090669, −2.42578313508821444493941527897, −2.10289925661687989317292007330, −1.97656073834855277041345530489, −1.36323688074543694122391554296, −1.26427182328738732777850530892, −0.988677189165820810713725797577, 0, 0, 0, 0.988677189165820810713725797577, 1.26427182328738732777850530892, 1.36323688074543694122391554296, 1.97656073834855277041345530489, 2.10289925661687989317292007330, 2.42578313508821444493941527897, 3.09696383709189731672943090669, 3.10859223609433760355684432072, 3.17818212786065565609748409028, 3.84233490312395565961706473238, 4.06546464417264798662282347153, 4.10395983870123998653375327020, 4.76887402181781235003506978590, 4.81565279272108814951564343679, 4.82192485556139645333758822505, 5.24243689373481727927031955730, 5.50429815028034712654363405230, 5.51136156742285234112041417663, 5.84811302662305641369194610051, 6.14357637020982794866056435722, 6.17149092093856526155808040517, 6.67446114524337539152497094972, 6.88427270806825451833729780774, 6.95687309128270076397110134511, 7.30123541572636076832801252673

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