Properties

Label 6-639e3-71.70-c0e3-0-0
Degree $6$
Conductor $260917119$
Sign $1$
Analytic cond. $0.0324320$
Root an. cond. $0.564714$
Motivic weight $0$
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  + 2-s + 5-s + 10-s − 19-s + 29-s − 37-s − 38-s − 43-s + 3·49-s + 58-s − 3·71-s − 73-s − 74-s − 79-s + 83-s − 86-s + 89-s − 95-s + 3·98-s + 101-s − 103-s − 6·107-s − 109-s + 3·121-s + 127-s − 128-s + 131-s + ⋯
L(s)  = 1  + 2-s + 5-s + 10-s − 19-s + 29-s − 37-s − 38-s − 43-s + 3·49-s + 58-s − 3·71-s − 73-s − 74-s − 79-s + 83-s − 86-s + 89-s − 95-s + 3·98-s + 101-s − 103-s − 6·107-s − 109-s + 3·121-s + 127-s − 128-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 71^{3}\)
Sign: $1$
Analytic conductor: \(0.0324320\)
Root analytic conductor: \(0.564714\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{639} (496, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{6} \cdot 71^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330590273\)
\(L(\frac12)\) \(\approx\) \(1.330590273\)
\(L(1)\) 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)$
bad3 \( 1 \)
71$C_1$ \( ( 1 + T )^{3} \)
good2$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
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$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
43$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
83$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
89$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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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

−9.833420775873518817936336720201, −9.170241421950823689049236844172, −9.146777271039715882952195593566, −8.836934329516558101851519555806, −8.486252079886083166878407002878, −8.264653899015990838034506744415, −7.84966148329591903118429176352, −7.53479988776860593174990930549, −7.01065521827902659158644931458, −6.96469621423864126178713937736, −6.57025284206264928971284169702, −6.17296074601218658421437252310, −5.94983306036910194011291288850, −5.44912832043962092066970112486, −5.43620883403123476947121773662, −5.03167812273964799938864423182, −4.43567647761754511569812735999, −4.28360347967664439419897041164, −4.18420663369478119619908585192, −3.55182342283842946945061968422, −3.08842507256238758207915794763, −2.69641103570264554325485886409, −2.28751464686665026535167388429, −1.77073079791246240609274844577, −1.26584969481661945262580678838, 1.26584969481661945262580678838, 1.77073079791246240609274844577, 2.28751464686665026535167388429, 2.69641103570264554325485886409, 3.08842507256238758207915794763, 3.55182342283842946945061968422, 4.18420663369478119619908585192, 4.28360347967664439419897041164, 4.43567647761754511569812735999, 5.03167812273964799938864423182, 5.43620883403123476947121773662, 5.44912832043962092066970112486, 5.94983306036910194011291288850, 6.17296074601218658421437252310, 6.57025284206264928971284169702, 6.96469621423864126178713937736, 7.01065521827902659158644931458, 7.53479988776860593174990930549, 7.84966148329591903118429176352, 8.264653899015990838034506744415, 8.486252079886083166878407002878, 8.836934329516558101851519555806, 9.146777271039715882952195593566, 9.170241421950823689049236844172, 9.833420775873518817936336720201

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