Properties

Label 6-1305e3-1.1-c1e3-0-2
Degree $6$
Conductor $2222447625$
Sign $-1$
Analytic cond. $1131.52$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·4-s − 3·5-s + 4·7-s + 2·8-s + 3·10-s − 2·11-s − 2·13-s − 4·14-s + 16-s + 4·17-s − 10·19-s + 6·20-s + 2·22-s − 16·23-s + 6·25-s + 2·26-s − 8·28-s + 3·29-s − 14·31-s + 32-s − 4·34-s − 12·35-s − 8·37-s + 10·38-s − 6·40-s + 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s − 1.34·5-s + 1.51·7-s + 0.707·8-s + 0.948·10-s − 0.603·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 2.29·19-s + 1.34·20-s + 0.426·22-s − 3.33·23-s + 6/5·25-s + 0.392·26-s − 1.51·28-s + 0.557·29-s − 2.51·31-s + 0.176·32-s − 0.685·34-s − 2.02·35-s − 1.31·37-s + 1.62·38-s − 0.948·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 5^{3} \cdot 29^{3}\)
Sign: $-1$
Analytic conductor: \(1131.52\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{6} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
29$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 52 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 25 T^{2} + 48 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 10 T + 85 T^{2} + 400 T^{3} + 85 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 16 T + 145 T^{2} + 36 p T^{3} + 145 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 14 T + 153 T^{2} + 944 T^{3} + 153 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 T + 87 T^{2} + 500 T^{3} + 87 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 39 T^{2} - 396 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T - 3 T^{2} - 176 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 14 T + 201 T^{2} + 1392 T^{3} + 201 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 155 T^{2} + 628 T^{3} + 155 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 75 T^{2} + 516 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 28 T + 453 T^{2} - 4468 T^{3} + 453 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 28 T + 389 T^{2} + 3704 T^{3} + 389 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 16 T + 119 T^{2} + 636 T^{3} + 119 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 149 T^{2} + 488 T^{3} + 149 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 1844 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 10 T + 279 T^{2} - 1740 T^{3} + 279 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 223 T^{2} - 1628 T^{3} + 223 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.985366969508367006331166122664, −8.499775554627435530103670311208, −8.474388164329219564535853373882, −8.167293485534335815797443050122, −8.027986117108595340978255802960, −7.71532560013958808765987318010, −7.58528448555998200914355214806, −7.35468412640114275331255431259, −6.84901370512058031546813207168, −6.53196820676728138265894050184, −6.17831254979941936347578402990, −5.80385595673914163595518467365, −5.71773564171144200719753467143, −4.97634793852087336703707632075, −4.90430784615762206092874395938, −4.83419416263156387978111592624, −4.19989824034651920284450046856, −4.14563899500907841908789549299, −3.91819856904807039536980891951, −3.30021799707193155491695692432, −3.23904655753008815214552442450, −2.40619006319160489572682357322, −2.12298168780212601923935844999, −1.54394993686830996169221983832, −1.50019452928248440997371526910, 0, 0, 0, 1.50019452928248440997371526910, 1.54394993686830996169221983832, 2.12298168780212601923935844999, 2.40619006319160489572682357322, 3.23904655753008815214552442450, 3.30021799707193155491695692432, 3.91819856904807039536980891951, 4.14563899500907841908789549299, 4.19989824034651920284450046856, 4.83419416263156387978111592624, 4.90430784615762206092874395938, 4.97634793852087336703707632075, 5.71773564171144200719753467143, 5.80385595673914163595518467365, 6.17831254979941936347578402990, 6.53196820676728138265894050184, 6.84901370512058031546813207168, 7.35468412640114275331255431259, 7.58528448555998200914355214806, 7.71532560013958808765987318010, 8.027986117108595340978255802960, 8.167293485534335815797443050122, 8.474388164329219564535853373882, 8.499775554627435530103670311208, 8.985366969508367006331166122664

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