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Properties

Label 6-431e3-431.430-c0e3-0-0
Degree $6$
Conductor $80062991$
Sign $1$
Analytic cond. $0.00995183$
Root an. cond. $0.463785$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s + 6·11-s + 15-s − 19-s − 6·22-s − 23-s − 29-s − 30-s − 6·33-s + 38-s − 41-s + 46-s + 3·49-s − 53-s − 6·55-s + 57-s + 58-s − 59-s − 61-s + 6·66-s + 69-s + 82-s + 87-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s + 6·11-s + 15-s − 19-s − 6·22-s − 23-s − 29-s − 30-s − 6·33-s + 38-s − 41-s + 46-s + 3·49-s − 53-s − 6·55-s + 57-s + 58-s − 59-s − 61-s + 6·66-s + 69-s + 82-s + 87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80062991 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80062991 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(80062991\)    =    \(431^{3}\)
Sign: $1$
Analytic conductor: \(0.00995183\)
Root analytic conductor: \(0.463785\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{431} (430, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 80062991,\ (\ :0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2229949714\)
\(L(\frac12)\) \(\approx\) \(0.2229949714\)
\(L(1)\) 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)$
bad431$C_1$ \( ( 1 - T )^{3} \)
good2$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
3$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
5$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_1$ \( ( 1 - T )^{6} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
19$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
23$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
29$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
53$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
59$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
61$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
97$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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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

−9.950207284205735111517235572170, −9.806850504849358835341516641413, −9.431382109769527729679870040274, −9.200620306338118211942704328064, −8.965703322197701109812993718759, −8.762861074425388030506493418900, −8.726611206171500724846871665549, −7.975740418490166470665667726491, −7.916432586930642498200672366598, −7.15155842429582796733241333464, −6.99697122923954466430808933502, −6.79738973199921560514862087173, −6.31790254194168164270036254696, −6.24798536474062340609840084125, −5.80794320174755592041415851401, −5.65044247565605566269180548385, −4.71072816950501530034895440899, −4.39876310194177161529601183170, −3.96976686128144615970864300945, −3.95465751271839231881092563948, −3.69157923564326064752331865616, −3.08646167109651913768329269533, −1.78247962675410925792230700719, −1.72928132395672597688784596139, −0.948072856839569317356302017586, 0.948072856839569317356302017586, 1.72928132395672597688784596139, 1.78247962675410925792230700719, 3.08646167109651913768329269533, 3.69157923564326064752331865616, 3.95465751271839231881092563948, 3.96976686128144615970864300945, 4.39876310194177161529601183170, 4.71072816950501530034895440899, 5.65044247565605566269180548385, 5.80794320174755592041415851401, 6.24798536474062340609840084125, 6.31790254194168164270036254696, 6.79738973199921560514862087173, 6.99697122923954466430808933502, 7.15155842429582796733241333464, 7.916432586930642498200672366598, 7.975740418490166470665667726491, 8.726611206171500724846871665549, 8.762861074425388030506493418900, 8.965703322197701109812993718759, 9.200620306338118211942704328064, 9.431382109769527729679870040274, 9.806850504849358835341516641413, 9.950207284205735111517235572170

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