Properties

Label 6-5070e3-1.1-c1e3-0-15
Degree $6$
Conductor $130323843000$
Sign $-1$
Analytic cond. $66352.1$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·3-s + 6·4-s + 3·5-s + 9·6-s − 10·7-s + 10·8-s + 6·9-s + 9·10-s − 8·11-s + 18·12-s − 30·14-s + 9·15-s + 15·16-s − 3·17-s + 18·18-s − 11·19-s + 18·20-s − 30·21-s − 24·22-s − 9·23-s + 30·24-s + 6·25-s + 10·27-s − 60·28-s + 27·30-s − 16·31-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s − 3.77·7-s + 3.53·8-s + 2·9-s + 2.84·10-s − 2.41·11-s + 5.19·12-s − 8.01·14-s + 2.32·15-s + 15/4·16-s − 0.727·17-s + 4.24·18-s − 2.52·19-s + 4.02·20-s − 6.54·21-s − 5.11·22-s − 1.87·23-s + 6.12·24-s + 6/5·25-s + 1.92·27-s − 11.3·28-s + 4.92·30-s − 2.87·31-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(66352.1\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
13 \( 1 \)
good7$A_4\times C_2$ \( 1 + 10 T + 52 T^{2} + 169 T^{3} + 52 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 8 T + 52 T^{2} + 189 T^{3} + 52 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 3 T + 5 T^{2} - 37 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 11 T + 53 T^{2} + 207 T^{3} + 53 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 9 T + 89 T^{2} + 427 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 24 T^{2} + 189 T^{3} + 24 p T^{4} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 16 T + 176 T^{2} + 1131 T^{3} + 176 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 10 T + 100 T^{2} + 517 T^{3} + 100 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 5 T + 87 T^{2} + 243 T^{3} + 87 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 3 T + 69 T^{2} + 385 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 3 T + 32 T^{2} - 277 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 17 T + 239 T^{2} + 1873 T^{3} + 239 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 11 T + 61 T^{2} + 219 T^{3} + 61 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 11 T + 193 T^{2} - 1313 T^{3} + 193 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 15 T + 213 T^{2} + 2009 T^{3} + 213 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
71$C_6$ \( 1 + 13 T + 183 T^{2} + 1833 T^{3} + 183 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - T - 21 T^{2} - 523 T^{3} - 21 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 3 T + 219 T^{2} + 447 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 12 T + 276 T^{2} + 1965 T^{3} + 276 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + T + 27 T^{2} - 831 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 2 T^{2} + 1397 T^{3} + 2 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.70408856446282039613889483990, −7.06411478265656074611572668885, −7.00408385427208481878939318674, −6.97126371523921316965019368431, −6.45111850754828813814990771475, −6.37568683958646331445120167253, −6.26499084696559944939049775778, −5.93641621171084126108905948038, −5.71909774909923964997920423971, −5.59946615019631795524472713991, −5.10402846570306680895657443090, −4.95699448450105013017182098975, −4.63116354171318905171812099676, −4.25980614360396930513062209504, −3.97904425336180202653208494975, −3.84137696285683360116665147451, −3.37766818113423552493457463966, −3.29526043631412132794423861778, −3.11804013131617215011432415306, −2.80378400877552721134890513426, −2.57317974669386789834925724601, −2.37439636495750090827273779390, −1.90722467113310526116393530741, −1.76248174778056944704193522496, −1.68852553460712449846146356513, 0, 0, 0, 1.68852553460712449846146356513, 1.76248174778056944704193522496, 1.90722467113310526116393530741, 2.37439636495750090827273779390, 2.57317974669386789834925724601, 2.80378400877552721134890513426, 3.11804013131617215011432415306, 3.29526043631412132794423861778, 3.37766818113423552493457463966, 3.84137696285683360116665147451, 3.97904425336180202653208494975, 4.25980614360396930513062209504, 4.63116354171318905171812099676, 4.95699448450105013017182098975, 5.10402846570306680895657443090, 5.59946615019631795524472713991, 5.71909774909923964997920423971, 5.93641621171084126108905948038, 6.26499084696559944939049775778, 6.37568683958646331445120167253, 6.45111850754828813814990771475, 6.97126371523921316965019368431, 7.00408385427208481878939318674, 7.06411478265656074611572668885, 7.70408856446282039613889483990

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