Properties

Label 6-405e3-1.1-c1e3-0-1
Degree $6$
Conductor $66430125$
Sign $1$
Analytic cond. $33.8217$
Root an. cond. $1.79831$
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  + 2-s + 3·5-s + 5·7-s + 3·10-s − 2·11-s + 4·13-s + 5·14-s − 16-s − 2·17-s + 4·19-s − 2·22-s + 3·23-s + 6·25-s + 4·26-s − 7·29-s + 8·31-s + 32-s − 2·34-s + 15·35-s + 6·37-s + 4·38-s − 13·41-s + 10·43-s + 3·46-s + 13·47-s + 49-s + 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s + 1.88·7-s + 0.948·10-s − 0.603·11-s + 1.10·13-s + 1.33·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.426·22-s + 0.625·23-s + 6/5·25-s + 0.784·26-s − 1.29·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 2.53·35-s + 0.986·37-s + 0.648·38-s − 2.03·41-s + 1.52·43-s + 0.442·46-s + 1.89·47-s + 1/7·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(66430125\)    =    \(3^{12} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(33.8217\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 66430125,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.040295690\)
\(L(\frac12)\) \(\approx\) \(5.040295690\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 5 T + 24 T^{2} - 67 T^{3} + 24 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + 35 T^{2} - 100 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 43 T^{2} + 56 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 3 T + 36 T^{2} - 21 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 2 p T^{2} + 355 T^{3} + 2 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 T + 33 T^{2} - 28 T^{3} + 33 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 99 T^{2} - 440 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 13 T + 142 T^{2} + 1069 T^{3} + 142 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 125 T^{2} - 856 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 13 T + 130 T^{2} - 853 T^{3} + 130 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 2 T + 139 T^{2} + 188 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T + 157 T^{2} + 212 T^{3} + 157 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 146 T^{2} - 193 T^{3} + 146 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 11 T + 162 T^{2} - 967 T^{3} + 162 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 10 T + 121 T^{2} + 712 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 155 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 153 T^{2} - 340 T^{3} + 153 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 15 T + 276 T^{2} + 2409 T^{3} + 276 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{3} \)
97$S_4\times C_2$ \( 1 + 18 T + 255 T^{2} + 2188 T^{3} + 255 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.23480782368015375175878622132, −9.699823814832895875174538969106, −9.392007957938456349797394014402, −9.257898121091855577726697204488, −8.617418563560361078466997369064, −8.603754952090573020173189531993, −8.281294856092912784424661004141, −7.78788256924353841025213093669, −7.65759607553751048416286278335, −7.03024273615491346463579841902, −7.00662119091092111598756542699, −6.28996164367483920774021968805, −6.16259807828959280219140478250, −5.54882013337785530863251845106, −5.51489292126511371491334709313, −5.23647463857818594601923063419, −4.55802444124845438928070684110, −4.48588607705119477716971697997, −4.29549291790146852028184952437, −3.37794355889215962104022456635, −3.17689899663705654627478208061, −2.38904308721300866982192686263, −2.23768245130265216385360926189, −1.38882093900943339040678353578, −1.21556430091835107025965573141, 1.21556430091835107025965573141, 1.38882093900943339040678353578, 2.23768245130265216385360926189, 2.38904308721300866982192686263, 3.17689899663705654627478208061, 3.37794355889215962104022456635, 4.29549291790146852028184952437, 4.48588607705119477716971697997, 4.55802444124845438928070684110, 5.23647463857818594601923063419, 5.51489292126511371491334709313, 5.54882013337785530863251845106, 6.16259807828959280219140478250, 6.28996164367483920774021968805, 7.00662119091092111598756542699, 7.03024273615491346463579841902, 7.65759607553751048416286278335, 7.78788256924353841025213093669, 8.281294856092912784424661004141, 8.603754952090573020173189531993, 8.617418563560361078466997369064, 9.257898121091855577726697204488, 9.392007957938456349797394014402, 9.699823814832895875174538969106, 10.23480782368015375175878622132

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