↑

Properties

Label 6-7942e3-1.1-c1e3-0-4
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

Learn more

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

\(L(1)\) \(\approx\) \(10.60461742\)
\(L(\frac12)\) \(\approx\) \(10.60461742\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 + 7 T + 214 T^{2} + 1227 T^{3} + 214 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.08805376829720276072796552814, −6.57230042390083161339429276196, −6.54270568574609973523151707768, −6.26450297222693722293088503567, −5.95021474418252463308943061820, −5.81252042426385396411941206224, −5.78445740073034063111739593672, −5.31992487604067035024713048689, −5.21859963961237801189411434149, −5.15448153621402214571052914819, −4.57120876651843869021698867760, −4.39722753097854635109886410570, −3.88636227186945849096951520428, −3.69552790413134398461240679271, −3.23783716707658810683821636807, −3.21589769080157814693368638075, −2.73969590443232533073312325214, −2.43206996464149772898703911886, −2.09407586469684325297821880253, −1.84611149477437802557070812135, −1.64510028011017743133170084595, −1.40396863931343866054051107548, −1.15667824241882436617112041397, −0.897719385913623566261293385454, −0.70620242318555125419011210041, 0.70620242318555125419011210041, 0.897719385913623566261293385454, 1.15667824241882436617112041397, 1.40396863931343866054051107548, 1.64510028011017743133170084595, 1.84611149477437802557070812135, 2.09407586469684325297821880253, 2.43206996464149772898703911886, 2.73969590443232533073312325214, 3.21589769080157814693368638075, 3.23783716707658810683821636807, 3.69552790413134398461240679271, 3.88636227186945849096951520428, 4.39722753097854635109886410570, 4.57120876651843869021698867760, 5.15448153621402214571052914819, 5.21859963961237801189411434149, 5.31992487604067035024713048689, 5.78445740073034063111739593672, 5.81252042426385396411941206224, 5.95021474418252463308943061820, 6.26450297222693722293088503567, 6.54270568574609973523151707768, 6.57230042390083161339429276196, 7.08805376829720276072796552814

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