Properties

Label 6-9280e3-1.1-c1e3-0-19
Degree $6$
Conductor $799178752000$
Sign $-1$
Analytic cond. $406888.$
Root an. cond. $8.60820$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 5^{3} \cdot 29^{3}\)
Sign: $-1$
Analytic conductor: \(406888.\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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

−6.85214545954407054549543402590, −6.75010693772042824110380954734, −6.60397224241894150322015858376, −6.47337066935274313755188895415, −6.10746239297793858504428590864, −5.98340533049131080702978745831, −5.91642843499371259951119453733, −5.59954103387503264860837286202, −5.49605422879285073542749374396, −5.15013783006961292462910627882, −4.79219603773129172167632555893, −4.73226245976432125516138012707, −4.42479689717374452642519478690, −4.04342492670660226349525627893, −3.81036317667390359180291114002, −3.62219220649379881599343472504, −3.40541851745542218855135336226, −3.11178995129076700574348379467, −2.91860865222623346989770033247, −2.17907675538213825529720609473, −2.00163096986397404555742743156, −1.65383045433746832654869897321, −1.54907339498591225861231479249, −1.38213314874475586651765752389, −1.24064574430798319532842339280, 0, 0, 0, 1.24064574430798319532842339280, 1.38213314874475586651765752389, 1.54907339498591225861231479249, 1.65383045433746832654869897321, 2.00163096986397404555742743156, 2.17907675538213825529720609473, 2.91860865222623346989770033247, 3.11178995129076700574348379467, 3.40541851745542218855135336226, 3.62219220649379881599343472504, 3.81036317667390359180291114002, 4.04342492670660226349525627893, 4.42479689717374452642519478690, 4.73226245976432125516138012707, 4.79219603773129172167632555893, 5.15013783006961292462910627882, 5.49605422879285073542749374396, 5.59954103387503264860837286202, 5.91642843499371259951119453733, 5.98340533049131080702978745831, 6.10746239297793858504428590864, 6.47337066935274313755188895415, 6.60397224241894150322015858376, 6.75010693772042824110380954734, 6.85214545954407054549543402590

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