Properties

Label 6-760e3-1.1-c1e3-0-0
Degree $6$
Conductor $438976000$
Sign $1$
Analytic cond. $223.497$
Root an. cond. $2.46345$
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-s + 3·5-s − 7-s − 2·9-s + 5·13-s − 3·15-s + 5·17-s + 3·19-s + 21-s − 23-s + 6·25-s + 3·27-s + 17·29-s + 2·31-s − 3·35-s + 8·37-s − 5·39-s + 6·41-s − 10·43-s − 6·45-s + 4·47-s − 8·49-s − 5·51-s + 5·53-s − 3·57-s + 15·59-s + 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.377·7-s − 2/3·9-s + 1.38·13-s − 0.774·15-s + 1.21·17-s + 0.688·19-s + 0.218·21-s − 0.208·23-s + 6/5·25-s + 0.577·27-s + 3.15·29-s + 0.359·31-s − 0.507·35-s + 1.31·37-s − 0.800·39-s + 0.937·41-s − 1.52·43-s − 0.894·45-s + 0.583·47-s − 8/7·49-s − 0.700·51-s + 0.686·53-s − 0.397·57-s + 1.95·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(223.497\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.117580238\)
\(L(\frac12)\) \(\approx\) \(3.117580238\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + T + 9 T^{2} - 2 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 - 5 T + 41 T^{2} - 126 T^{3} + 41 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 5 T + 47 T^{2} - 166 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + T + 37 T^{2} - 18 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 17 T + 171 T^{2} - 1110 T^{3} + 171 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} + 4 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 61 T^{2} - 504 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 117 T^{2} - 582 T^{3} + 117 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 15 T + 149 T^{2} - 986 T^{3} + 149 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 23 T^{2} + 650 T^{3} + 23 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 4 T + 21 T^{2} - 456 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 3 T + 119 T^{2} - 10 p T^{3} + 119 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 213 T^{2} - 284 T^{3} + 213 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 233 T^{2} + 1628 T^{3} + 233 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 187 T^{2} + 1072 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.339608443846956485332514463948, −8.815038998151942066650079151664, −8.681384638026606490842714315350, −8.389629143354915464958881966870, −8.021247988585065163288621818349, −8.009450292800877235090905566131, −7.43257497693855931358395546361, −6.88725203173550604459093765087, −6.77035964981026072585441356101, −6.46868095677231596224256272830, −6.07836165716150442674164725842, −6.07433500239069738366994744695, −5.62868764048575192137593915729, −5.24576155296133299179287079208, −5.14529422628463535960614763253, −4.79315188530252213002848303014, −4.09783949361175612796706099188, −3.95142913392650859996777779527, −3.46187234887498844093351198676, −2.90155047877099722007122839638, −2.62516524298711256247397858528, −2.55099828486234539448364061704, −1.49818681625352366274665120402, −1.20344395864318515258100406463, −0.75797977384676029689490623979, 0.75797977384676029689490623979, 1.20344395864318515258100406463, 1.49818681625352366274665120402, 2.55099828486234539448364061704, 2.62516524298711256247397858528, 2.90155047877099722007122839638, 3.46187234887498844093351198676, 3.95142913392650859996777779527, 4.09783949361175612796706099188, 4.79315188530252213002848303014, 5.14529422628463535960614763253, 5.24576155296133299179287079208, 5.62868764048575192137593915729, 6.07433500239069738366994744695, 6.07836165716150442674164725842, 6.46868095677231596224256272830, 6.77035964981026072585441356101, 6.88725203173550604459093765087, 7.43257497693855931358395546361, 8.009450292800877235090905566131, 8.021247988585065163288621818349, 8.389629143354915464958881966870, 8.681384638026606490842714315350, 8.815038998151942066650079151664, 9.339608443846956485332514463948

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