Properties

Label 6-525e3-1.1-c1e3-0-1
Degree $6$
Conductor $144703125$
Sign $1$
Analytic cond. $73.6731$
Root an. cond. $2.04747$
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  − 2-s + 3·3-s − 3·6-s + 3·7-s − 2·8-s + 6·9-s + 6·11-s + 6·13-s − 3·14-s + 3·16-s − 6·18-s + 6·19-s + 9·21-s − 6·22-s − 4·23-s − 6·24-s − 6·26-s + 10·27-s + 2·29-s + 2·31-s − 3·32-s + 18·33-s + 4·37-s − 6·38-s + 18·39-s + 2·41-s − 9·42-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.13·7-s − 0.707·8-s + 2·9-s + 1.80·11-s + 1.66·13-s − 0.801·14-s + 3/4·16-s − 1.41·18-s + 1.37·19-s + 1.96·21-s − 1.27·22-s − 0.834·23-s − 1.22·24-s − 1.17·26-s + 1.92·27-s + 0.371·29-s + 0.359·31-s − 0.530·32-s + 3.13·33-s + 0.657·37-s − 0.973·38-s + 2.88·39-s + 0.312·41-s − 1.38·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(73.6731\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.154458227\)
\(L(\frac12)\) \(\approx\) \(4.154458227\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.488955697269998902726127343570, −9.311266427577489277333430127999, −9.240701171692835591998690599415, −8.709674419243524329560312130844, −8.486807367448386961998377543363, −8.260314135259506262863829014208, −8.095944194541780680855303610191, −7.68234739127162575410322514300, −7.59127026891978329060420902585, −6.82472178482904034539484641598, −6.79374606396769027039006230987, −6.24284072893515137486339851277, −6.22924475067329073486711598694, −5.63385722833276316572021417302, −5.02522262844140395814578506502, −4.98967227666984831662205492010, −4.15816000723335585044553795505, −4.05716625139708045112469583870, −3.66481207120444030082643463708, −3.31967611255563260991031607198, −3.00963324380550823526572834362, −2.33657418047064760612703806245, −1.85573080646300267740297235574, −1.22392924948101114786523316269, −1.14951268463002916512455010277, 1.14951268463002916512455010277, 1.22392924948101114786523316269, 1.85573080646300267740297235574, 2.33657418047064760612703806245, 3.00963324380550823526572834362, 3.31967611255563260991031607198, 3.66481207120444030082643463708, 4.05716625139708045112469583870, 4.15816000723335585044553795505, 4.98967227666984831662205492010, 5.02522262844140395814578506502, 5.63385722833276316572021417302, 6.22924475067329073486711598694, 6.24284072893515137486339851277, 6.79374606396769027039006230987, 6.82472178482904034539484641598, 7.59127026891978329060420902585, 7.68234739127162575410322514300, 8.095944194541780680855303610191, 8.260314135259506262863829014208, 8.486807367448386961998377543363, 8.709674419243524329560312130844, 9.240701171692835591998690599415, 9.311266427577489277333430127999, 9.488955697269998902726127343570

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