Properties

Label 6-825e3-1.1-c1e3-0-4
Degree $6$
Conductor $561515625$
Sign $1$
Analytic cond. $285.886$
Root an. cond. $2.56664$
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 − 2·4-s + 3·6-s + 4·7-s − 2·8-s + 6·9-s + 3·11-s − 6·12-s + 2·13-s + 4·14-s + 16-s + 6·18-s − 6·19-s + 12·21-s + 3·22-s + 12·23-s − 6·24-s + 2·26-s + 10·27-s − 8·28-s + 8·29-s − 8·31-s − 32-s + 9·33-s − 12·36-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 4-s + 1.22·6-s + 1.51·7-s − 0.707·8-s + 2·9-s + 0.904·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.41·18-s − 1.37·19-s + 2.61·21-s + 0.639·22-s + 2.50·23-s − 1.22·24-s + 0.392·26-s + 1.92·27-s − 1.51·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s + 1.56·33-s − 2·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(285.886\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.887783193\)
\(L(\frac12)\) \(\approx\) \(8.887783193\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 52 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 48 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 23 T^{2} + 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 - 8 T + 71 T^{2} - 304 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 8 T + 11 T^{2} + 272 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 884 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 16 T + 189 T^{2} - 1536 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 191 T^{2} - 1712 T^{3} + 191 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 22 T + 291 T^{2} + 2692 T^{3} + 291 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 185 T^{2} - 1544 T^{3} + 185 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 117 T^{2} + 760 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 175 T^{2} + 1072 T^{3} + 175 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 10 T + 25 T^{2} - 140 T^{3} + 25 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 189 T^{2} + 1056 T^{3} + 189 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 255 T^{2} - 2684 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 16 T + 131 T^{2} - 672 T^{3} + 131 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.104203974436054120229060692702, −8.765401354852331276024806660120, −8.614624849082227082319612573810, −8.567757934604622706684121475401, −7.87795856184446085183920192508, −7.79470942502315420280141410663, −7.41889197249961132116921462587, −7.26608016874994634906267709187, −6.74837658191249750479556754827, −6.66843397542123878391877571256, −5.97651217865648722128749755082, −5.73901189515565344498359566074, −5.52329144866696438507384362168, −4.73497171417031631695587691727, −4.70395705559365580448502452741, −4.59649657740377297783744434237, −4.16173390288034934150778132376, −3.80115792904191761314367007499, −3.74425489410673704852611651654, −2.96594741682823080045916528160, −2.79104211969081013802893288998, −2.29615279519136901163764750232, −1.87826704779462494268479947443, −1.10772739208189699857074096964, −1.05530810322569037039931007084, 1.05530810322569037039931007084, 1.10772739208189699857074096964, 1.87826704779462494268479947443, 2.29615279519136901163764750232, 2.79104211969081013802893288998, 2.96594741682823080045916528160, 3.74425489410673704852611651654, 3.80115792904191761314367007499, 4.16173390288034934150778132376, 4.59649657740377297783744434237, 4.70395705559365580448502452741, 4.73497171417031631695587691727, 5.52329144866696438507384362168, 5.73901189515565344498359566074, 5.97651217865648722128749755082, 6.66843397542123878391877571256, 6.74837658191249750479556754827, 7.26608016874994634906267709187, 7.41889197249961132116921462587, 7.79470942502315420280141410663, 7.87795856184446085183920192508, 8.567757934604622706684121475401, 8.614624849082227082319612573810, 8.765401354852331276024806660120, 9.104203974436054120229060692702

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