Properties

Label 6-6840e3-1.1-c1e3-0-4
Degree $6$
Conductor $320013504000$
Sign $1$
Analytic cond. $162929.$
Root an. cond. $7.39037$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 6·11-s + 4·13-s − 12·17-s + 3·19-s + 4·23-s + 6·25-s − 6·31-s − 4·37-s − 2·41-s + 2·43-s − 4·47-s − 7·49-s + 10·53-s + 18·55-s − 2·59-s + 14·61-s − 12·65-s + 12·67-s + 4·71-s + 18·73-s − 2·79-s + 6·83-s + 36·85-s − 10·89-s − 9·95-s + 32·97-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.80·11-s + 1.10·13-s − 2.91·17-s + 0.688·19-s + 0.834·23-s + 6/5·25-s − 1.07·31-s − 0.657·37-s − 0.312·41-s + 0.304·43-s − 0.583·47-s − 49-s + 1.37·53-s + 2.42·55-s − 0.260·59-s + 1.79·61-s − 1.48·65-s + 1.46·67-s + 0.474·71-s + 2.10·73-s − 0.225·79-s + 0.658·83-s + 3.90·85-s − 1.05·89-s − 0.923·95-s + 3.24·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(162929.\)
Root analytic conductor: \(7.39037\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.453961322\)
\(L(\frac12)\) \(\approx\) \(2.453961322\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + p T^{2} + 12 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 31 T^{2} + 100 T^{3} + 31 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + T^{2} + 60 T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
23$S_4\times C_2$ \( 1 - 4 T + 25 T^{2} - 152 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 73 T^{2} + 12 T^{3} + 73 p T^{4} + p^{3} T^{6} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 + 4 T + 33 T^{2} + 476 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + p T^{2} - 180 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T + 87 T^{2} - 252 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 97 T^{2} + 344 T^{3} + 97 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 143 T^{2} - 964 T^{3} + 143 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T + 33 T^{2} - 452 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 1564 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 - 4 T + 45 T^{2} - 1208 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
79$S_4\times C_2$ \( 1 + 2 T + 45 T^{2} - 708 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 205 T^{2} - 796 T^{3} + 205 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 10 T + 217 T^{2} + 1828 T^{3} + 217 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 32 T + 549 T^{2} - 6244 T^{3} + 549 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.15662901913473291712496317273, −6.78150661206004317132619650485, −6.70948586618996500422522391632, −6.38942274588562971702310613518, −6.22242969175495081563243247493, −5.77432144885595697517727758774, −5.62348746928534938394869660433, −5.15393651104344861543434160364, −5.13611453727594716769207104007, −4.97525233285085837104077820219, −4.53945119716778137561510076577, −4.48819527642475406629501672974, −4.10689326046625969273427254169, −3.62185793346839368812218234722, −3.60727152806803622941416807047, −3.58284454442487163696595953688, −2.97391527214936521217148963772, −2.74324356271774964766129324211, −2.55463245235526571176693829221, −2.06383609804152082296926841603, −1.78396171876200716098222588106, −1.77249081805890786946030085571, −0.74834249765321032478226912845, −0.55982416789551528294546411912, −0.47729104476777837586869306108, 0.47729104476777837586869306108, 0.55982416789551528294546411912, 0.74834249765321032478226912845, 1.77249081805890786946030085571, 1.78396171876200716098222588106, 2.06383609804152082296926841603, 2.55463245235526571176693829221, 2.74324356271774964766129324211, 2.97391527214936521217148963772, 3.58284454442487163696595953688, 3.60727152806803622941416807047, 3.62185793346839368812218234722, 4.10689326046625969273427254169, 4.48819527642475406629501672974, 4.53945119716778137561510076577, 4.97525233285085837104077820219, 5.13611453727594716769207104007, 5.15393651104344861543434160364, 5.62348746928534938394869660433, 5.77432144885595697517727758774, 6.22242969175495081563243247493, 6.38942274588562971702310613518, 6.70948586618996500422522391632, 6.78150661206004317132619650485, 7.15662901913473291712496317273

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