Properties

Label 6-6930e3-1.1-c1e3-0-2
Degree $6$
Conductor $332812557000$
Sign $1$
Analytic cond. $169445.$
Root an. cond. $7.43883$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s + 3·5-s + 3·7-s − 10·8-s − 9·10-s + 3·11-s − 9·14-s + 15·16-s + 6·17-s − 6·19-s + 18·20-s − 9·22-s + 6·23-s + 6·25-s + 18·28-s + 6·29-s − 21·32-s − 18·34-s + 9·35-s − 6·37-s + 18·38-s − 30·40-s + 6·41-s − 6·43-s + 18·44-s − 18·46-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s + 1.34·5-s + 1.13·7-s − 3.53·8-s − 2.84·10-s + 0.904·11-s − 2.40·14-s + 15/4·16-s + 1.45·17-s − 1.37·19-s + 4.02·20-s − 1.91·22-s + 1.25·23-s + 6/5·25-s + 3.40·28-s + 1.11·29-s − 3.71·32-s − 3.08·34-s + 1.52·35-s − 0.986·37-s + 2.91·38-s − 4.74·40-s + 0.937·41-s − 0.914·43-s + 2.71·44-s − 2.65·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(169445.\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.042520696\)
\(L(\frac12)\) \(\approx\) \(5.042520696\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
good13$A_4\times C_2$ \( 1 + 27 T^{2} - 8 T^{3} + 27 p T^{4} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 6 T + 3 p T^{2} - 180 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 6 T + 21 T^{2} + 4 p T^{3} + 21 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 6 T - 3 T^{2} + 180 T^{3} - 3 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 6 T + 51 T^{2} - 324 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 45 T^{2} + 64 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 436 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 6 T + 87 T^{2} - 468 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 6 T + 93 T^{2} + 364 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
53$A_4\times C_2$ \( 1 - 6 T + 123 T^{2} - 612 T^{3} + 123 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 1224 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
67$A_4\times C_2$ \( 1 - 12 T + 93 T^{2} - 896 T^{3} + 93 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 6 T + 33 T^{2} + 36 T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 6 T + 183 T^{2} + 724 T^{3} + 183 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 45 T^{2} - 512 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 105 T^{2} - 576 T^{3} + 105 p T^{4} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 12 T + 87 T^{2} - 432 T^{3} + 87 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 159 T^{2} - 1460 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19875615957025183864430365092, −6.77158758952053823195426772354, −6.65736375018455513234698414203, −6.47822036137780368777781861831, −6.24681560808909870081090276666, −5.86528822037508913396607550244, −5.75988713873773532899181444454, −5.36724751294884522783624321661, −5.34284418741521565314573579884, −5.08726516474830540981999275756, −4.45493730148384649497486967251, −4.40977347241790630803480861853, −4.31692720720859001491031399357, −3.53070941439505275352262703978, −3.43986847550685079072255867010, −3.34767078732276831058188571422, −2.65085672423893477198451842824, −2.52583099221584571913346368885, −2.37688289244150345560057126046, −1.85497288173185631919011707290, −1.68165888970459513691517705221, −1.58145068022609358610088140695, −0.851286187519780566946491845247, −0.792997566518613641270351565529, −0.68089779953372271872870901000, 0.68089779953372271872870901000, 0.792997566518613641270351565529, 0.851286187519780566946491845247, 1.58145068022609358610088140695, 1.68165888970459513691517705221, 1.85497288173185631919011707290, 2.37688289244150345560057126046, 2.52583099221584571913346368885, 2.65085672423893477198451842824, 3.34767078732276831058188571422, 3.43986847550685079072255867010, 3.53070941439505275352262703978, 4.31692720720859001491031399357, 4.40977347241790630803480861853, 4.45493730148384649497486967251, 5.08726516474830540981999275756, 5.34284418741521565314573579884, 5.36724751294884522783624321661, 5.75988713873773532899181444454, 5.86528822037508913396607550244, 6.24681560808909870081090276666, 6.47822036137780368777781861831, 6.65736375018455513234698414203, 6.77158758952053823195426772354, 7.19875615957025183864430365092

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