Properties

Label 6-3040e3-1.1-c1e3-0-7
Degree $6$
Conductor $28094464000$
Sign $-1$
Analytic cond. $14303.8$
Root an. cond. $4.92691$
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  + 3·5-s + 2·7-s − 5·9-s − 2·11-s − 4·13-s − 10·17-s − 3·19-s − 6·23-s + 6·25-s + 2·27-s − 14·29-s + 10·31-s + 6·35-s − 2·37-s − 20·41-s − 10·43-s − 15·45-s − 6·47-s − 9·49-s − 10·53-s − 6·55-s + 6·59-s − 18·61-s − 10·63-s − 12·65-s + 2·67-s + 6·71-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.755·7-s − 5/3·9-s − 0.603·11-s − 1.10·13-s − 2.42·17-s − 0.688·19-s − 1.25·23-s + 6/5·25-s + 0.384·27-s − 2.59·29-s + 1.79·31-s + 1.01·35-s − 0.328·37-s − 3.12·41-s − 1.52·43-s − 2.23·45-s − 0.875·47-s − 9/7·49-s − 1.37·53-s − 0.809·55-s + 0.781·59-s − 2.30·61-s − 1.25·63-s − 1.48·65-s + 0.244·67-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{15} \cdot 5^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(14303.8\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{15} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 + 2 T + 29 T^{2} + 40 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 41 T^{2} + 102 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 10 T + 63 T^{2} + 300 T^{3} + 63 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 53 T^{2} + 176 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 14 T + 99 T^{2} + 516 T^{3} + 99 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 10 T + 105 T^{2} - 580 T^{3} + 105 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 20 T + 219 T^{2} + 1608 T^{3} + 219 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 + 6 T + 125 T^{2} + 464 T^{3} + 125 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 10 T + 133 T^{2} + 726 T^{3} + 133 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T + 69 T^{2} - 492 T^{3} + 69 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 18 T + 243 T^{2} + 2144 T^{3} + 243 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 2 T + 137 T^{2} - 354 T^{3} + 137 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 65 T^{2} - 1148 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 2 T + 71 T^{2} - 588 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 18 T + 225 T^{2} - 2324 T^{3} + 225 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 2 T + 157 T^{2} - 600 T^{3} + 157 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 183 T^{2} + 588 T^{3} + 183 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 93 T^{2} - 1218 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.281309550728136847840602518886, −7.85895850653421738583582495265, −7.68924718839764349945328810183, −7.50835988738111158461644995467, −6.84900100655481773393968362940, −6.71570698018752142952680001006, −6.62243099448065715812332948873, −6.23274062207919122270060707047, −6.10446903960095174923496010787, −6.02393363414844966553224566024, −5.25594802133391614190117883015, −5.19124563299928196332641251004, −5.13173687180765345443604641226, −4.82757493047801878413756294850, −4.65299312853105614982684568665, −4.27532332749531487902873461989, −3.72182836916553050431711000108, −3.38440485260519177232508606994, −3.37857705295460401275568762224, −2.67575030018766442103152315214, −2.46326495125272572352308101429, −2.26806311634257313015407009517, −2.02557223296903036683766298670, −1.53576914066282992121202055297, −1.46651765978515138806234245328, 0, 0, 0, 1.46651765978515138806234245328, 1.53576914066282992121202055297, 2.02557223296903036683766298670, 2.26806311634257313015407009517, 2.46326495125272572352308101429, 2.67575030018766442103152315214, 3.37857705295460401275568762224, 3.38440485260519177232508606994, 3.72182836916553050431711000108, 4.27532332749531487902873461989, 4.65299312853105614982684568665, 4.82757493047801878413756294850, 5.13173687180765345443604641226, 5.19124563299928196332641251004, 5.25594802133391614190117883015, 6.02393363414844966553224566024, 6.10446903960095174923496010787, 6.23274062207919122270060707047, 6.62243099448065715812332948873, 6.71570698018752142952680001006, 6.84900100655481773393968362940, 7.50835988738111158461644995467, 7.68924718839764349945328810183, 7.85895850653421738583582495265, 8.281309550728136847840602518886

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