Properties

Label 6-7800e3-1.1-c1e3-0-6
Degree $6$
Conductor $474552000000$
Sign $1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
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  + 3·3-s + 7-s + 6·9-s + 5·11-s − 3·13-s + 9·17-s − 2·19-s + 3·21-s − 4·23-s + 10·27-s − 5·29-s + 5·31-s + 15·33-s − 6·37-s − 9·39-s + 13·47-s + 3·49-s + 27·51-s − 53-s − 6·57-s + 13·59-s + 11·61-s + 6·63-s + 25·67-s − 12·69-s + 2·71-s + 2·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.832·13-s + 2.18·17-s − 0.458·19-s + 0.654·21-s − 0.834·23-s + 1.92·27-s − 0.928·29-s + 0.898·31-s + 2.61·33-s − 0.986·37-s − 1.44·39-s + 1.89·47-s + 3/7·49-s + 3.78·51-s − 0.137·53-s − 0.794·57-s + 1.69·59-s + 1.40·61-s + 0.755·63-s + 3.05·67-s − 1.44·69-s + 0.237·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.76957652\)
\(L(\frac12)\) \(\approx\) \(17.76957652\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - T - 2 T^{2} + 19 T^{3} - 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 28 T^{2} - 107 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 2 T + 21 T^{2} + 136 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 37 T^{2} + 52 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 42 T^{2} + 89 T^{3} + 42 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 88 T^{2} - 307 T^{3} + 88 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 6 T + 87 T^{2} + 424 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 87 T^{2} + 44 T^{3} + 87 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 93 T^{2} + 44 T^{3} + 93 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 13 T + 122 T^{2} - 861 T^{3} + 122 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + T + 122 T^{2} + 9 T^{3} + 122 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 13 T + 158 T^{2} - 1173 T^{3} + 158 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 11 T + 170 T^{2} - 1079 T^{3} + 170 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 25 T + 396 T^{2} - 3803 T^{3} + 396 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 2 T + 73 T^{2} + 352 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 2 T + 79 T^{2} + 344 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 249 T^{2} - 1860 T^{3} + 249 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 15 T + 246 T^{2} + 2027 T^{3} + 246 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 16 T + 299 T^{2} - 2832 T^{3} + 299 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 14 T + 263 T^{2} - 2180 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
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

−6.95854364719864713013783604718, −6.75193209661976176695186451325, −6.62751692463367441628695004932, −6.35461242470062830841247372333, −6.01687707204312962909138970265, −5.73930512848878012683760905168, −5.50547225896383492599794177341, −5.22890129171987079901899199530, −5.10168040988772283660109414464, −4.77684653503503663264839759301, −4.35388704016881001649707589494, −4.26304463214415203097323793349, −3.84004931377888263640819524964, −3.64160493747148429484824713898, −3.61559699676243755537690634684, −3.49732200851230148692659161025, −2.81438974161304000629079081042, −2.75190589013385557680916195905, −2.40779945037751932210840022539, −1.98663758868016661306937390962, −1.93914655913954452848599334817, −1.67436161630326493790896512940, −0.998398824874256283170876879894, −0.888104340252599814559270953916, −0.58770634195269297456990820581, 0.58770634195269297456990820581, 0.888104340252599814559270953916, 0.998398824874256283170876879894, 1.67436161630326493790896512940, 1.93914655913954452848599334817, 1.98663758868016661306937390962, 2.40779945037751932210840022539, 2.75190589013385557680916195905, 2.81438974161304000629079081042, 3.49732200851230148692659161025, 3.61559699676243755537690634684, 3.64160493747148429484824713898, 3.84004931377888263640819524964, 4.26304463214415203097323793349, 4.35388704016881001649707589494, 4.77684653503503663264839759301, 5.10168040988772283660109414464, 5.22890129171987079901899199530, 5.50547225896383492599794177341, 5.73930512848878012683760905168, 6.01687707204312962909138970265, 6.35461242470062830841247372333, 6.62751692463367441628695004932, 6.75193209661976176695186451325, 6.95854364719864713013783604718

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