Properties

Label 6-7942e3-1.1-c1e3-0-6
Degree $6$
Conductor $500944540888$
Sign $1$
Analytic cond. $255047.$
Root an. cond. $7.96349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s + 6·5-s + 6·7-s + 10·8-s − 9-s + 18·10-s + 3·11-s − 7·13-s + 18·14-s + 15·16-s + 10·17-s − 3·18-s + 36·20-s + 9·22-s + 12·23-s + 9·25-s − 21·26-s + 2·27-s + 36·28-s + 3·29-s − 2·31-s + 21·32-s + 30·34-s + 36·35-s − 6·36-s − 4·37-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 2.68·5-s + 2.26·7-s + 3.53·8-s − 1/3·9-s + 5.69·10-s + 0.904·11-s − 1.94·13-s + 4.81·14-s + 15/4·16-s + 2.42·17-s − 0.707·18-s + 8.04·20-s + 1.91·22-s + 2.50·23-s + 9/5·25-s − 4.11·26-s + 0.384·27-s + 6.80·28-s + 0.557·29-s − 0.359·31-s + 3.71·32-s + 5.14·34-s + 6.08·35-s − 36-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 11^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(255047.\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 11^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(89.39130270\)
\(L(\frac12)\) \(\approx\) \(89.39130270\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
19 \( 1 \)
good3$S_4\times C_2$ \( 1 + T^{2} - 2 T^{3} + p T^{4} + p^{3} T^{6} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
7$S_4\times C_2$ \( 1 - 6 T + 25 T^{2} - 78 T^{3} + 25 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 7 T + 34 T^{2} + 111 T^{3} + 34 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 10 T + 61 T^{2} - 290 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 12 T + 85 T^{2} - 472 T^{3} + 85 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 + 2 T + 53 T^{2} + 8 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 41 T^{2} + 446 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 121 T^{2} + 766 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 5 T + 102 T^{2} - 349 T^{3} + 102 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 11 T + 100 T^{2} + 533 T^{3} + 100 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 31 T^{2} + 128 T^{3} + 31 p T^{4} + p^{3} T^{6} \)
59$C_2$ \( ( 1 + p T^{2} )^{3} \)
61$S_4\times C_2$ \( 1 - 11 T + 158 T^{2} - 967 T^{3} + 158 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 129 T^{2} + 54 T^{3} + 129 p T^{4} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 5 T + 186 T^{2} - 629 T^{3} + 186 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 8 T + 205 T^{2} - 1066 T^{3} + 205 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 177 T^{2} + 758 T^{3} + 177 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 154 T^{2} - 1367 T^{3} + 154 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + T + 174 T^{2} + 221 T^{3} + 174 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 7 T + 214 T^{2} - 1227 T^{3} + 214 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.85101162669711019841057013453, −6.53082107467626594513236548548, −6.36222735048332424203851301407, −6.25033800371169634573143725316, −5.67785334544387720613862663172, −5.53122408590765501699423491338, −5.48573716885447014276025524975, −5.22692778574317429611077468893, −5.21777414882370772995893214755, −4.89671192840050223684744721605, −4.73620179271286978403083221513, −4.35964899898857972189051263559, −4.34858909980298133229597318757, −3.56924868512559938229508658175, −3.55612164909932728432017826456, −3.39268599143974655770292923586, −2.78696612678683951857599673305, −2.65705631587251714138939010817, −2.63798853810012910525230211139, −1.95714779601577247844752340026, −1.79076878487283522700893932619, −1.64420719392920401411414189235, −1.63779376005893889972726705672, −0.950235893761683292244513349617, −0.76800017200529102891408070510, 0.76800017200529102891408070510, 0.950235893761683292244513349617, 1.63779376005893889972726705672, 1.64420719392920401411414189235, 1.79076878487283522700893932619, 1.95714779601577247844752340026, 2.63798853810012910525230211139, 2.65705631587251714138939010817, 2.78696612678683951857599673305, 3.39268599143974655770292923586, 3.55612164909932728432017826456, 3.56924868512559938229508658175, 4.34858909980298133229597318757, 4.35964899898857972189051263559, 4.73620179271286978403083221513, 4.89671192840050223684744721605, 5.21777414882370772995893214755, 5.22692778574317429611077468893, 5.48573716885447014276025524975, 5.53122408590765501699423491338, 5.67785334544387720613862663172, 6.25033800371169634573143725316, 6.36222735048332424203851301407, 6.53082107467626594513236548548, 6.85101162669711019841057013453

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