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Properties

Label 6-546e3-1.1-c7e3-0-1
Degree $6$
Conductor $162771336$
Sign $-1$
Analytic cond. $4.96190\times 10^{6}$
Root an. cond. $13.0599$
Motivic weight $7$
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  + 24·2-s + 81·3-s + 384·4-s − 378·5-s + 1.94e3·6-s − 1.02e3·7-s + 5.12e3·8-s + 4.37e3·9-s − 9.07e3·10-s − 6.06e3·11-s + 3.11e4·12-s + 6.59e3·13-s − 2.46e4·14-s − 3.06e4·15-s + 6.14e4·16-s − 1.22e4·17-s + 1.04e5·18-s − 4.83e3·19-s − 1.45e5·20-s − 8.33e4·21-s − 1.45e5·22-s + 1.11e5·23-s + 4.14e5·24-s − 2.47e4·25-s + 1.58e5·26-s + 1.96e5·27-s − 3.95e5·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s − 1.35·5-s + 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s − 2.86·10-s − 1.37·11-s + 5.19·12-s + 0.832·13-s − 2.40·14-s − 2.34·15-s + 15/4·16-s − 0.603·17-s + 4.24·18-s − 0.161·19-s − 4.05·20-s − 1.96·21-s − 2.91·22-s + 1.91·23-s + 6.12·24-s − 0.317·25-s + 1.76·26-s + 1.92·27-s − 3.40·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(4.96190\times 10^{6}\)
Root analytic conductor: \(13.0599\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3} ,\ ( \ : 7/2, 7/2, 7/2 ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{3} \)
3$C_1$ \( ( 1 - p^{3} T )^{3} \)
7$C_1$ \( ( 1 + p^{3} T )^{3} \)
13$C_1$ \( ( 1 - p^{3} T )^{3} \)
good5$S_4\times C_2$ \( 1 + 378 T + 33534 p T^{2} + 2460962 p^{2} T^{3} + 33534 p^{8} T^{4} + 378 p^{14} T^{5} + p^{21} T^{6} \)
11$S_4\times C_2$ \( 1 + 6069 T + 57737715 T^{2} + 225516090214 T^{3} + 57737715 p^{7} T^{4} + 6069 p^{14} T^{5} + p^{21} T^{6} \)
17$S_4\times C_2$ \( 1 + 12231 T + 1015192989 T^{2} + 441040510390 p T^{3} + 1015192989 p^{7} T^{4} + 12231 p^{14} T^{5} + p^{21} T^{6} \)
19$S_4\times C_2$ \( 1 + 4836 T + 752922264 T^{2} + 35017972159972 T^{3} + 752922264 p^{7} T^{4} + 4836 p^{14} T^{5} + p^{21} T^{6} \)
23$S_4\times C_2$ \( 1 - 111606 T + 13698692358 T^{2} - 789331279915484 T^{3} + 13698692358 p^{7} T^{4} - 111606 p^{14} T^{5} + p^{21} T^{6} \)
29$S_4\times C_2$ \( 1 - 105414 T + 53303773566 T^{2} - 3610494153306790 T^{3} + 53303773566 p^{7} T^{4} - 105414 p^{14} T^{5} + p^{21} T^{6} \)
31$S_4\times C_2$ \( 1 + 263061 T + 1981515915 p T^{2} + 9620069398446870 T^{3} + 1981515915 p^{8} T^{4} + 263061 p^{14} T^{5} + p^{21} T^{6} \)
37$S_4\times C_2$ \( 1 + 15837 p T + 322648348677 T^{2} + 96319812043814042 T^{3} + 322648348677 p^{7} T^{4} + 15837 p^{15} T^{5} + p^{21} T^{6} \)
41$S_4\times C_2$ \( 1 + 649608 T + 658820658759 T^{2} + 251009633606388400 T^{3} + 658820658759 p^{7} T^{4} + 649608 p^{14} T^{5} + p^{21} T^{6} \)
43$S_4\times C_2$ \( 1 + 76182 T + 786301609674 T^{2} + 42739995151961600 T^{3} + 786301609674 p^{7} T^{4} + 76182 p^{14} T^{5} + p^{21} T^{6} \)
47$S_4\times C_2$ \( 1 + 1269219 T + 1809049905309 T^{2} + 1226963184489914394 T^{3} + 1809049905309 p^{7} T^{4} + 1269219 p^{14} T^{5} + p^{21} T^{6} \)
53$S_4\times C_2$ \( 1 - 326511 T + 1095139727751 T^{2} - 1097458164564560514 T^{3} + 1095139727751 p^{7} T^{4} - 326511 p^{14} T^{5} + p^{21} T^{6} \)
59$S_4\times C_2$ \( 1 + 3434316 T + 9433035834681 T^{2} + 15369494893998268488 T^{3} + 9433035834681 p^{7} T^{4} + 3434316 p^{14} T^{5} + p^{21} T^{6} \)
61$S_4\times C_2$ \( 1 + 1450497 T + 8283604093131 T^{2} + 8643456086000399014 T^{3} + 8283604093131 p^{7} T^{4} + 1450497 p^{14} T^{5} + p^{21} T^{6} \)
67$S_4\times C_2$ \( 1 + 1302372 T + 11799899385981 T^{2} + 10414858400928054536 T^{3} + 11799899385981 p^{7} T^{4} + 1302372 p^{14} T^{5} + p^{21} T^{6} \)
71$S_4\times C_2$ \( 1 + 5186076 T + 17007117555813 T^{2} + 34747260910145466632 T^{3} + 17007117555813 p^{7} T^{4} + 5186076 p^{14} T^{5} + p^{21} T^{6} \)
73$S_4\times C_2$ \( 1 + 3662940 T + 17923949464536 T^{2} + 47671317275867890926 T^{3} + 17923949464536 p^{7} T^{4} + 3662940 p^{14} T^{5} + p^{21} T^{6} \)
79$S_4\times C_2$ \( 1 + 2950251 T + 47925511200855 T^{2} + \)\(11\!\cdots\!86\)\( T^{3} + 47925511200855 p^{7} T^{4} + 2950251 p^{14} T^{5} + p^{21} T^{6} \)
83$S_4\times C_2$ \( 1 + 13650225 T + 122886468691635 T^{2} + \)\(76\!\cdots\!14\)\( T^{3} + 122886468691635 p^{7} T^{4} + 13650225 p^{14} T^{5} + p^{21} T^{6} \)
89$S_4\times C_2$ \( 1 - 6379533 T + 99908233712583 T^{2} - \)\(48\!\cdots\!34\)\( T^{3} + 99908233712583 p^{7} T^{4} - 6379533 p^{14} T^{5} + p^{21} T^{6} \)
97$S_4\times C_2$ \( 1 + 18032235 T + 213688600971087 T^{2} + \)\(22\!\cdots\!90\)\( T^{3} + 213688600971087 p^{7} T^{4} + 18032235 p^{14} T^{5} + p^{21} T^{6} \)
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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

−9.040212484041951801600950000475, −8.449193198012071299110452451822, −8.199889576328755469249161465774, −8.108034760016223827127449384811, −7.47492244394749194675770620516, −7.41922560293322537938545467774, −7.21957056677134774828160055594, −6.71337277206907552260292489329, −6.48125001826533880540862554125, −6.43080238028431462636425759172, −5.55489830330273292310830605482, −5.43384502174015893541523997049, −5.27029241939908995757640831361, −4.55656696253067250059487721233, −4.34466889432099229667444067442, −4.31998217902809258816368635319, −3.61551179716829046495661806448, −3.40465492227486525421696827968, −3.33737723972870870097326535661, −2.98486393280944505775422126308, −2.65725745715169734434240022721, −2.44897325136209904415397654924, −1.63360987625152307421371673852, −1.55053206047715501296059239104, −1.25071996214478156010945813700, 0, 0, 0, 1.25071996214478156010945813700, 1.55053206047715501296059239104, 1.63360987625152307421371673852, 2.44897325136209904415397654924, 2.65725745715169734434240022721, 2.98486393280944505775422126308, 3.33737723972870870097326535661, 3.40465492227486525421696827968, 3.61551179716829046495661806448, 4.31998217902809258816368635319, 4.34466889432099229667444067442, 4.55656696253067250059487721233, 5.27029241939908995757640831361, 5.43384502174015893541523997049, 5.55489830330273292310830605482, 6.43080238028431462636425759172, 6.48125001826533880540862554125, 6.71337277206907552260292489329, 7.21957056677134774828160055594, 7.41922560293322537938545467774, 7.47492244394749194675770620516, 8.108034760016223827127449384811, 8.199889576328755469249161465774, 8.449193198012071299110452451822, 9.040212484041951801600950000475

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