Properties

Label 6-675e3-1.1-c3e3-0-3
Degree $6$
Conductor $307546875$
Sign $-1$
Analytic cond. $63169.8$
Root an. cond. $6.31080$
Motivic weight $3$
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 − 44·7-s − 9·8-s − 38·11-s − 28·13-s + 44·14-s + 11·16-s − 19·17-s + 187·19-s + 38·22-s − 81·23-s + 28·26-s − 160·29-s + 227·31-s − 58·32-s + 19·34-s − 78·37-s − 187·38-s + 338·41-s − 22·43-s + 81·46-s − 472·47-s + 355·49-s + 521·53-s + 396·56-s + 160·58-s − 140·59-s + ⋯
L(s)  = 1  − 0.353·2-s − 2.37·7-s − 0.397·8-s − 1.04·11-s − 0.597·13-s + 0.839·14-s + 0.171·16-s − 0.271·17-s + 2.25·19-s + 0.368·22-s − 0.734·23-s + 0.211·26-s − 1.02·29-s + 1.31·31-s − 0.320·32-s + 0.0958·34-s − 0.346·37-s − 0.798·38-s + 1.28·41-s − 0.0780·43-s + 0.259·46-s − 1.46·47-s + 1.03·49-s + 1.35·53-s + 0.944·56-s + 0.362·58-s − 0.308·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{9} \cdot 5^{6}\)
Sign: $-1$
Analytic conductor: \(63169.8\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{675} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{9} \cdot 5^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2$S_4\times C_2$ \( 1 + T + T^{2} + 5 p T^{3} + p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 44 T + 1581 T^{2} + 31984 T^{3} + 1581 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 38 T + 1381 T^{2} + 17876 T^{3} + 1381 p^{3} T^{4} + 38 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 28 T + 155 p T^{2} + 22912 T^{3} + 155 p^{4} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 19 T + 3262 T^{2} - 367193 T^{3} + 3262 p^{3} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 187 T + 24164 T^{2} - 2039395 T^{3} + 24164 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 81 T + 13200 T^{2} - 72927 T^{3} + 13200 p^{3} T^{4} + 81 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 160 T + 25399 T^{2} - 88280 T^{3} + 25399 p^{3} T^{4} + 160 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 227 T + 71400 T^{2} - 13771435 T^{3} + 71400 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 78 T + 52035 T^{2} - 5735212 T^{3} + 52035 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 338 T + 163951 T^{2} - 34473956 T^{3} + 163951 p^{3} T^{4} - 338 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 22 T + 78605 T^{2} - 14966252 T^{3} + 78605 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 472 T + 365677 T^{2} + 97725712 T^{3} + 365677 p^{3} T^{4} + 472 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 521 T + 508018 T^{2} - 154190045 T^{3} + 508018 p^{3} T^{4} - 521 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 140 T + 449689 T^{2} + 23374640 T^{3} + 449689 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 878 T + 978237 T^{2} + 516844828 T^{3} + 978237 p^{3} T^{4} + 878 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 602 T + 490081 T^{2} - 150373964 T^{3} + 490081 p^{3} T^{4} - 602 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 1294 T + 1071143 T^{2} + 602684716 T^{3} + 1071143 p^{3} T^{4} + 1294 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 629 T + 1576176 T^{2} - 622253365 T^{3} + 1576176 p^{3} T^{4} - 629 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 1287 T + 1349484 T^{2} + 1125375327 T^{3} + 1349484 p^{3} T^{4} + 1287 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 2154 T + 3172479 T^{2} + 3111332052 T^{3} + 3172479 p^{3} T^{4} + 2154 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1392 T + 3281955 T^{2} + 2604477152 T^{3} + 3281955 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6} \)
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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

−9.523971702247407299411588211658, −9.106309948557660823742550032545, −8.972635155990737467013127521931, −8.653895123171495304630688470262, −8.135713212416500450137870138357, −7.84571209656769414474315739518, −7.71156658833001926476947611776, −7.20308636640991476464467929224, −7.14478822281319723859118112369, −6.62203805076771838156904969524, −6.54315020216551930978633735327, −5.92056388232809856038357131665, −5.91313362681760704864163283228, −5.53229775656126475448198560204, −5.12452559077187558901512581917, −4.88989995733891607714094941092, −4.29309479827068826488386546601, −3.94257719189974779877219500002, −3.51855424481931352723135829536, −3.15231960446786648172493864248, −2.95478437740811664520517183444, −2.57677249517291400594606789658, −2.28720332740287806150101776029, −1.32324874297173100205937929991, −1.13238179255917717196505540243, 0, 0, 0, 1.13238179255917717196505540243, 1.32324874297173100205937929991, 2.28720332740287806150101776029, 2.57677249517291400594606789658, 2.95478437740811664520517183444, 3.15231960446786648172493864248, 3.51855424481931352723135829536, 3.94257719189974779877219500002, 4.29309479827068826488386546601, 4.88989995733891607714094941092, 5.12452559077187558901512581917, 5.53229775656126475448198560204, 5.91313362681760704864163283228, 5.92056388232809856038357131665, 6.54315020216551930978633735327, 6.62203805076771838156904969524, 7.14478822281319723859118112369, 7.20308636640991476464467929224, 7.71156658833001926476947611776, 7.84571209656769414474315739518, 8.135713212416500450137870138357, 8.653895123171495304630688470262, 8.972635155990737467013127521931, 9.106309948557660823742550032545, 9.523971702247407299411588211658

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