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Properties

Label 6-6080e3-1.1-c1e3-0-8
Degree $6$
Conductor $224755712000$
Sign $-1$
Analytic cond. $114430.$
Root an. cond. $6.96771$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 5^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(114430.\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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

−7.45062617082185847348722461963, −7.02214667517474305107724535911, −6.87372448087659206438027077263, −6.87062197794712497716988621669, −6.65749293835552996455107771651, −6.32602882600973618583049577412, −6.29446085310604421694659158399, −5.73917520843622185967264886913, −5.56982257594971124084840958331, −5.32780050558839463809726563319, −4.93138067550564493788895576881, −4.83447047834634339351434745677, −4.75338171429834403178384041009, −4.16985407856463795449464605056, −3.91434561742717163314605163597, −3.88111850941777763420051327814, −3.43781554497291866041327921215, −3.14585411157768778130130998453, −3.10786037568007634802939211279, −2.59269238084168374894823841245, −2.41143644903691165510635692484, −2.29483547525440025268823915940, −1.63140583847017557649148626786, −1.22886988820728869147530092022, −0.870921391643859194607947233119, 0, 0, 0, 0.870921391643859194607947233119, 1.22886988820728869147530092022, 1.63140583847017557649148626786, 2.29483547525440025268823915940, 2.41143644903691165510635692484, 2.59269238084168374894823841245, 3.10786037568007634802939211279, 3.14585411157768778130130998453, 3.43781554497291866041327921215, 3.88111850941777763420051327814, 3.91434561742717163314605163597, 4.16985407856463795449464605056, 4.75338171429834403178384041009, 4.83447047834634339351434745677, 4.93138067550564493788895576881, 5.32780050558839463809726563319, 5.56982257594971124084840958331, 5.73917520843622185967264886913, 6.29446085310604421694659158399, 6.32602882600973618583049577412, 6.65749293835552996455107771651, 6.87062197794712497716988621669, 6.87372448087659206438027077263, 7.02214667517474305107724535911, 7.45062617082185847348722461963

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