Properties

Label 6-7800e3-1.1-c1e3-0-9
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·7-s + 6·9-s + 3·11-s − 3·13-s − 17-s + 4·19-s + 15·21-s − 5·23-s − 10·27-s − 4·29-s + 10·31-s − 9·33-s − 5·37-s + 9·39-s + 5·41-s − 12·43-s − 2·47-s + 10·49-s + 3·51-s + 13·53-s − 12·57-s + 21·61-s − 30·63-s − 8·67-s + 15·69-s − 71-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.88·7-s + 2·9-s + 0.904·11-s − 0.832·13-s − 0.242·17-s + 0.917·19-s + 3.27·21-s − 1.04·23-s − 1.92·27-s − 0.742·29-s + 1.79·31-s − 1.56·33-s − 0.821·37-s + 1.44·39-s + 0.780·41-s − 1.82·43-s − 0.291·47-s + 10/7·49-s + 0.420·51-s + 1.78·53-s − 1.58·57-s + 2.68·61-s − 3.77·63-s − 0.977·67-s + 1.80·69-s − 0.118·71-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: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$A_4\times C_2$ \( 1 + 5 T + 15 T^{2} + 38 T^{3} + 15 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
11$C_6$ \( 1 - 3 T - 7 T^{2} + 62 T^{3} - 7 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + T + 37 T^{2} + 42 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 4 T + 5 T^{2} + 24 T^{3} + 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 5 T + 63 T^{2} + 198 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 4 T + 35 T^{2} + 56 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 10 T + 69 T^{2} - 364 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 5 T + 19 T^{2} - 178 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 5 T + 31 T^{2} + 138 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
47$A_4\times C_2$ \( 1 + 2 T + 85 T^{2} + 252 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 13 T + 115 T^{2} - 894 T^{3} + 115 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
59$C_2$ \( ( 1 + p T^{2} )^{3} \)
61$A_4\times C_2$ \( 1 - 21 T + 287 T^{2} - 2518 T^{3} + 287 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 8 T - 7 T^{2} - 336 T^{3} - 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + T + 113 T^{2} + 494 T^{3} + 113 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 28 T + 423 T^{2} + 4264 T^{3} + 423 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 21 T + 341 T^{2} + 3446 T^{3} + 341 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 4 T + 25 T^{2} + 1176 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 11 T + 35 T^{2} + 638 T^{3} + 35 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 25 T + 5 p T^{2} + 5322 T^{3} + 5 p^{2} T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19754059397209531726990316454, −6.93174518419641587488784183833, −6.83494729066855348160558386871, −6.65431305468119600928870934547, −6.18168776703387905654728020822, −6.08159930954048913171413715655, −6.05974869280620965801607588458, −5.78395954036553390677787135410, −5.35748191054468132901884935810, −5.32486880881121005663599234908, −4.99129813816770127634435054249, −4.74846549647423271260092969258, −4.44173500626481641236939759559, −4.06897090504468550859421121555, −3.99389288392653420459943815428, −3.88726345628404907882787314954, −3.25168627037502626658719082789, −3.21326304555492483364763456294, −2.97750195242884071040178092935, −2.47980075866146917381987099552, −2.25332611827170463168943344081, −1.94447194517144223528820569135, −1.37836944126421206270177940754, −1.06413965304453787467947673615, −1.00064592207440773781752775952, 0, 0, 0, 1.00064592207440773781752775952, 1.06413965304453787467947673615, 1.37836944126421206270177940754, 1.94447194517144223528820569135, 2.25332611827170463168943344081, 2.47980075866146917381987099552, 2.97750195242884071040178092935, 3.21326304555492483364763456294, 3.25168627037502626658719082789, 3.88726345628404907882787314954, 3.99389288392653420459943815428, 4.06897090504468550859421121555, 4.44173500626481641236939759559, 4.74846549647423271260092969258, 4.99129813816770127634435054249, 5.32486880881121005663599234908, 5.35748191054468132901884935810, 5.78395954036553390677787135410, 6.05974869280620965801607588458, 6.08159930954048913171413715655, 6.18168776703387905654728020822, 6.65431305468119600928870934547, 6.83494729066855348160558386871, 6.93174518419641587488784183833, 7.19754059397209531726990316454

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