Properties

Label 6-1425e3-1.1-c1e3-0-4
Degree $6$
Conductor $2893640625$
Sign $1$
Analytic cond. $1473.24$
Root an. cond. $3.37323$
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  + 3·3-s − 3·8-s + 6·9-s − 3·11-s + 6·13-s + 3·19-s − 3·23-s − 9·24-s + 10·27-s + 9·29-s + 9·31-s − 9·33-s + 12·37-s + 18·39-s + 24·43-s − 6·47-s − 12·49-s + 9·53-s + 9·57-s + 24·59-s − 9·61-s + 64-s + 9·67-s − 9·69-s + 6·71-s − 18·72-s − 3·73-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.06·8-s + 2·9-s − 0.904·11-s + 1.66·13-s + 0.688·19-s − 0.625·23-s − 1.83·24-s + 1.92·27-s + 1.67·29-s + 1.61·31-s − 1.56·33-s + 1.97·37-s + 2.88·39-s + 3.65·43-s − 0.875·47-s − 1.71·49-s + 1.23·53-s + 1.19·57-s + 3.12·59-s − 1.15·61-s + 1/8·64-s + 1.09·67-s − 1.08·69-s + 0.712·71-s − 2.12·72-s − 0.351·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(1473.24\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.453490267\)
\(L(\frac12)\) \(\approx\) \(7.453490267\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good2$D_{6}$ \( 1 + 3 T^{3} + p^{3} T^{6} \)
7$D_{6}$ \( 1 + 12 T^{2} + 4 T^{3} + 12 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 27 T^{2} + 54 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 27 T^{2} - 92 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + p T^{2} )^{3} \)
23$S_4\times C_2$ \( 1 + 3 T + 45 T^{2} + 150 T^{3} + 45 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 - 9 T + 3 p T^{2} - 542 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 704 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
41$D_{6}$ \( 1 + 426 T^{3} + p^{3} T^{6} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{3} \)
47$S_4\times C_2$ \( 1 + 6 T + 81 T^{2} + 660 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 9 T + 90 T^{2} - 885 T^{3} + 90 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 24 T + 360 T^{2} - 3276 T^{3} + 360 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 9 T + 174 T^{2} + 985 T^{3} + 174 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 9 T + 135 T^{2} - 1298 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 132 T^{2} - 330 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 3 T + 126 T^{2} + 535 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 15 T + 189 T^{2} - 1454 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 15 T + 315 T^{2} + 2574 T^{3} + 315 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 3 T + 234 T^{2} + 531 T^{3} + 234 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 12 T + 123 T^{2} + 880 T^{3} + 123 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.376242849134875061007593236510, −8.112223088602822480795813905441, −8.080166020208417316138290174897, −7.981433715685128735233749861301, −7.60583346852054660816995680653, −6.99872030586306958033732657513, −6.89144635828577464043125168307, −6.68840310874937769841187580499, −6.23497550084210427650172753437, −6.01854000434960821990958855695, −5.77288139647343232522822230986, −5.37947314189008662982103910911, −5.17071929849830422259778729572, −4.50086211635991439638389892579, −4.32664668896782373582464097133, −4.15952642739462073963337956073, −3.61991659231700699572599331651, −3.55509853241516110056284173074, −2.78556194172172236322488934756, −2.75590044579045009746241961286, −2.72431488935428205132416763006, −2.21476187028307508214873968040, −1.57432478050364414297587662814, −0.926181602410925988083071930296, −0.817368586730479305433292227350, 0.817368586730479305433292227350, 0.926181602410925988083071930296, 1.57432478050364414297587662814, 2.21476187028307508214873968040, 2.72431488935428205132416763006, 2.75590044579045009746241961286, 2.78556194172172236322488934756, 3.55509853241516110056284173074, 3.61991659231700699572599331651, 4.15952642739462073963337956073, 4.32664668896782373582464097133, 4.50086211635991439638389892579, 5.17071929849830422259778729572, 5.37947314189008662982103910911, 5.77288139647343232522822230986, 6.01854000434960821990958855695, 6.23497550084210427650172753437, 6.68840310874937769841187580499, 6.89144635828577464043125168307, 6.99872030586306958033732657513, 7.60583346852054660816995680653, 7.981433715685128735233749861301, 8.080166020208417316138290174897, 8.112223088602822480795813905441, 8.376242849134875061007593236510

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