Properties

Label 6-1080e3-1.1-c3e3-0-11
Degree $6$
Conductor $1259712000$
Sign $-1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
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  + 15·5-s − 27·11-s − 3·13-s − 15·17-s − 78·19-s − 105·23-s + 150·25-s − 117·29-s − 207·31-s − 120·37-s − 300·41-s − 483·43-s − 303·47-s − 522·49-s + 492·53-s − 405·55-s − 240·59-s − 444·61-s − 45·65-s − 522·67-s + 168·71-s − 876·73-s − 2.10e3·79-s + 42·83-s − 225·85-s + 2.26e3·89-s − 1.17e3·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.740·11-s − 0.0640·13-s − 0.214·17-s − 0.941·19-s − 0.951·23-s + 6/5·25-s − 0.749·29-s − 1.19·31-s − 0.533·37-s − 1.14·41-s − 1.71·43-s − 0.940·47-s − 1.52·49-s + 1.27·53-s − 0.992·55-s − 0.529·59-s − 0.931·61-s − 0.0858·65-s − 0.951·67-s + 0.280·71-s − 1.40·73-s − 2.99·79-s + 0.0555·83-s − 0.287·85-s + 2.70·89-s − 1.26·95-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 522 T^{2} - 2666 T^{3} + 522 p^{3} T^{4} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 27 T + 2721 T^{2} + 79918 T^{3} + 2721 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 4962 T^{2} + 32735 T^{3} + 4962 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 15 T + 5031 T^{2} - 273298 T^{3} + 5031 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 78 T + 9942 T^{2} + 303500 T^{3} + 9942 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 105 T + 25413 T^{2} + 2052802 T^{3} + 25413 p^{3} T^{4} + 105 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 117 T + 67683 T^{2} + 5431966 T^{3} + 67683 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 207 T + 69741 T^{2} + 8616098 T^{3} + 69741 p^{3} T^{4} + 207 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 120 T - 39264 T^{2} - 10421922 T^{3} - 39264 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 300 T + 82023 T^{2} + 31686600 T^{3} + 82023 p^{3} T^{4} + 300 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 483 T + 251961 T^{2} + 76432898 T^{3} + 251961 p^{3} T^{4} + 483 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 303 T + 166137 T^{2} + 65064754 T^{3} + 166137 p^{3} T^{4} + 303 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 492 T + 476259 T^{2} - 138372072 T^{3} + 476259 p^{3} T^{4} - 492 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 240 T + 213705 T^{2} - 18543136 T^{3} + 213705 p^{3} T^{4} + 240 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 444 T + 185052 T^{2} - 860902 T^{3} + 185052 p^{3} T^{4} + 444 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 522 T + 370530 T^{2} + 307525264 T^{3} + 370530 p^{3} T^{4} + 522 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 168 T - 41151 T^{2} + 401172816 T^{3} - 41151 p^{3} T^{4} - 168 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 12 p T + 1148628 T^{2} + 668051142 T^{3} + 1148628 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 2103 T + 2885448 T^{2} + 2370776439 T^{3} + 2885448 p^{3} T^{4} + 2103 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 42 T + 653901 T^{2} - 416514772 T^{3} + 653901 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 2268 T + 3125643 T^{2} - 3038479992 T^{3} + 3125643 p^{3} T^{4} - 2268 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1392 T + 3023340 T^{2} + 2460493802 T^{3} + 3023340 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.819671284804964907223357144890, −8.504651727910553825373027251328, −8.378880810674521467222831904237, −8.114452480360851587845004336251, −7.59938936184512779101892961852, −7.34783149105685143453484575132, −7.29787297828388830971350481095, −6.67665545388968059483684720342, −6.42431943506750949001302539055, −6.41928275435902914827362905465, −5.90637681931119857788960111012, −5.65293825069743491992370100570, −5.47383722609249902881616097631, −4.90863519105514332967960224071, −4.81984906132525224764758151887, −4.69041698136533586428660614100, −3.89024952802177831716447916998, −3.64841282267826420412917611486, −3.56627494681457311495608723035, −2.70724192828852963877299952642, −2.70257400147801841515595620050, −2.30890533200176278373009543196, −1.69037607003984394073699291536, −1.45161828626845281510298223159, −1.42832356651378652296097868180, 0, 0, 0, 1.42832356651378652296097868180, 1.45161828626845281510298223159, 1.69037607003984394073699291536, 2.30890533200176278373009543196, 2.70257400147801841515595620050, 2.70724192828852963877299952642, 3.56627494681457311495608723035, 3.64841282267826420412917611486, 3.89024952802177831716447916998, 4.69041698136533586428660614100, 4.81984906132525224764758151887, 4.90863519105514332967960224071, 5.47383722609249902881616097631, 5.65293825069743491992370100570, 5.90637681931119857788960111012, 6.41928275435902914827362905465, 6.42431943506750949001302539055, 6.67665545388968059483684720342, 7.29787297828388830971350481095, 7.34783149105685143453484575132, 7.59938936184512779101892961852, 8.114452480360851587845004336251, 8.378880810674521467222831904237, 8.504651727910553825373027251328, 8.819671284804964907223357144890

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