Properties

Label 6-3040e3-1.1-c1e3-0-6
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

−8.092503522388348298958147397219, −7.937628267191987964791619905102, −7.45134239399546305259248176562, −7.25434860183952808228317534126, −6.97041803190922164229463500413, −6.80623410885709461557086029279, −6.78824696016997509612977609771, −6.08518322118070209548618230151, −6.01656440300720097313071234779, −6.00825459924150429446555468115, −5.43933394021951906555524099045, −5.29125501088101962842728215891, −5.24886201715245999558166971704, −4.67744456902282165995443375809, −4.58730037381520938106215440954, −4.14685202017847989071547698720, −3.78459189862710463516581790233, −3.32300772674790813735121862622, −3.30604687578726080251411384349, −2.79115382641516411144824972807, −2.64568692671073515884323644331, −2.34203169077999906920532045271, −1.74188107202953454702611274082, −1.66676547351451847819791366123, −1.38232949222341698401447607469, 0, 0, 0, 1.38232949222341698401447607469, 1.66676547351451847819791366123, 1.74188107202953454702611274082, 2.34203169077999906920532045271, 2.64568692671073515884323644331, 2.79115382641516411144824972807, 3.30604687578726080251411384349, 3.32300772674790813735121862622, 3.78459189862710463516581790233, 4.14685202017847989071547698720, 4.58730037381520938106215440954, 4.67744456902282165995443375809, 5.24886201715245999558166971704, 5.29125501088101962842728215891, 5.43933394021951906555524099045, 6.00825459924150429446555468115, 6.01656440300720097313071234779, 6.08518322118070209548618230151, 6.78824696016997509612977609771, 6.80623410885709461557086029279, 6.97041803190922164229463500413, 7.25434860183952808228317534126, 7.45134239399546305259248176562, 7.937628267191987964791619905102, 8.092503522388348298958147397219

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