Properties

Label 6-2320e3-1.1-c1e3-0-7
Degree $6$
Conductor $12487168000$
Sign $-1$
Analytic cond. $6357.63$
Root an. cond. $4.30410$
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  + 2·3-s − 3·5-s + 2·7-s − 9-s − 8·11-s − 6·13-s − 6·15-s + 4·21-s − 14·23-s + 6·25-s − 4·27-s + 3·29-s − 12·31-s − 16·33-s − 6·35-s + 4·37-s − 12·39-s − 10·41-s + 10·43-s + 3·45-s − 18·47-s − 9·49-s + 10·53-s + 24·55-s − 4·59-s + 6·61-s − 2·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s + 0.755·7-s − 1/3·9-s − 2.41·11-s − 1.66·13-s − 1.54·15-s + 0.872·21-s − 2.91·23-s + 6/5·25-s − 0.769·27-s + 0.557·29-s − 2.15·31-s − 2.78·33-s − 1.01·35-s + 0.657·37-s − 1.92·39-s − 1.56·41-s + 1.52·43-s + 0.447·45-s − 2.62·47-s − 9/7·49-s + 1.37·53-s + 3.23·55-s − 0.520·59-s + 0.768·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 29^{3}\)
Sign: $-1$
Analytic conductor: \(6357.63\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
29$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 8 T + 49 T^{2} + 180 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 11 T^{2} + 76 T^{3} + 11 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 29 T^{2} - 52 T^{3} + 29 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 14 T + 129 T^{2} + 720 T^{3} + 129 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 12 T + 113 T^{2} + 748 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 71 T^{2} - 228 T^{3} + 71 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 157 T^{2} - 880 T^{3} + 157 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 18 T + 201 T^{2} + 1600 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 179 T^{2} - 1052 T^{3} + 179 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 129 T^{2} + 552 T^{3} + 129 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 179 T^{2} - 692 T^{3} + 179 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 10 T + 229 T^{2} - 1360 T^{3} + 229 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 24 T + 389 T^{2} + 3776 T^{3} + 389 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 4 T + 39 T^{2} - 524 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 8 T + 181 T^{2} + 1244 T^{3} + 181 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 2 T + 217 T^{2} - 384 T^{3} + 217 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 22 T + 391 T^{2} - 4116 T^{3} + 391 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 36 T + 639 T^{2} + 7436 T^{3} + 639 p T^{4} + 36 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.387990829881667670066576221086, −8.135819676395002133051709048490, −7.88002495231077656060814451814, −7.62456300728160561758620876105, −7.50038557886195642778481924766, −7.27152483053227450223493339523, −7.18446360201357078092142758448, −6.52055702165479018622212677102, −6.27837355755144583870355559572, −6.04341870009173345002313835651, −5.55625160752083595672875090853, −5.29522980375089032546163196992, −5.24853358501967100201659016714, −4.79716271025353039136340835981, −4.58566172672220683836171139399, −4.41116613196407847411868033492, −3.78053824157967615217411963831, −3.68478541654179143411143881818, −3.45396964382149648558793537038, −2.86256311432694956236547411841, −2.70050274549717620539678227744, −2.52079415511911958956344111553, −2.09601458715400040731379040194, −1.84271647601281067439300861706, −1.26525992624491365667413558009, 0, 0, 0, 1.26525992624491365667413558009, 1.84271647601281067439300861706, 2.09601458715400040731379040194, 2.52079415511911958956344111553, 2.70050274549717620539678227744, 2.86256311432694956236547411841, 3.45396964382149648558793537038, 3.68478541654179143411143881818, 3.78053824157967615217411963831, 4.41116613196407847411868033492, 4.58566172672220683836171139399, 4.79716271025353039136340835981, 5.24853358501967100201659016714, 5.29522980375089032546163196992, 5.55625160752083595672875090853, 6.04341870009173345002313835651, 6.27837355755144583870355559572, 6.52055702165479018622212677102, 7.18446360201357078092142758448, 7.27152483053227450223493339523, 7.50038557886195642778481924766, 7.62456300728160561758620876105, 7.88002495231077656060814451814, 8.135819676395002133051709048490, 8.387990829881667670066576221086

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