Properties

Label 6-91e6-1.1-c1e3-0-0
Degree $6$
Conductor $567869252041$
Sign $1$
Analytic cond. $289121.$
Root an. cond. $8.13167$
Motivic weight $1$
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  + 2·2-s + 4·3-s + 2·4-s − 5·5-s + 8·6-s + 2·8-s + 9·9-s − 10·10-s + 4·11-s + 8·12-s − 20·15-s + 4·17-s + 18·18-s − 7·19-s − 10·20-s + 8·22-s + 23-s + 8·24-s + 7·25-s + 18·27-s − 7·29-s − 40·30-s + 3·31-s + 16·33-s + 8·34-s + 18·36-s + 10·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s − 2.23·5-s + 3.26·6-s + 0.707·8-s + 3·9-s − 3.16·10-s + 1.20·11-s + 2.30·12-s − 5.16·15-s + 0.970·17-s + 4.24·18-s − 1.60·19-s − 2.23·20-s + 1.70·22-s + 0.208·23-s + 1.63·24-s + 7/5·25-s + 3.46·27-s − 1.29·29-s − 7.30·30-s + 0.538·31-s + 2.78·33-s + 1.37·34-s + 3·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(289121.\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8281} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 7^{6} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.311914051\)
\(L(\frac12)\) \(\approx\) \(1.311914051\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13 \( 1 \)
good2$S_4\times C_2$ \( 1 - p T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 - 4 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + p T + 18 T^{2} + 43 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 86 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} - 122 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 7 T + 54 T^{2} + 203 T^{3} + 54 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 28 T^{2} - 89 T^{3} + 28 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 52 T^{2} - 137 T^{3} + 52 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 119 T^{2} - 658 T^{3} + 119 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 6 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 17 T + 230 T^{2} + 1745 T^{3} + 230 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 13 T + 198 T^{2} - 1369 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 22 T + 321 T^{2} + 2848 T^{3} + 321 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 24 T + 343 T^{2} + 3152 T^{3} + 343 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 14 T + 165 T^{2} - 1228 T^{3} + 165 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 4 T + 169 T^{2} + 374 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 5 T - 831 T^{3} + 5 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - T + 168 T^{2} - 257 T^{3} + 168 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 23 T + 376 T^{2} + 4021 T^{3} + 376 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 11 T + 212 T^{2} - 1979 T^{3} + 212 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 3 T + 220 T^{2} - 575 T^{3} + 220 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.96789300546689287624223675114, −6.67850555915381894993069883876, −6.42311979002370115900379216214, −6.20595546160058381871778447327, −6.10706833429456187146494911629, −5.75759787049948060661816797671, −5.46107864327298834396471372858, −4.91370093632105664588851604055, −4.71088742390143150025999012365, −4.63666973568343889153864361843, −4.36657825506386897037955937032, −4.13563700530741479003316408972, −4.09636116675525963257431052194, −3.74432570402243959399499837831, −3.48028898817510986394662683005, −3.33418564749965143315614199732, −3.04160084244775062970196918873, −3.03330668137900794484602867880, −2.46467010183649451482424581321, −2.28516135905643779703159234275, −1.82741957222823032856560523558, −1.74829608143831410433394175499, −1.13986448135650014277351090630, −0.965119752509008547013541931495, −0.097311000460654536829109568775, 0.097311000460654536829109568775, 0.965119752509008547013541931495, 1.13986448135650014277351090630, 1.74829608143831410433394175499, 1.82741957222823032856560523558, 2.28516135905643779703159234275, 2.46467010183649451482424581321, 3.03330668137900794484602867880, 3.04160084244775062970196918873, 3.33418564749965143315614199732, 3.48028898817510986394662683005, 3.74432570402243959399499837831, 4.09636116675525963257431052194, 4.13563700530741479003316408972, 4.36657825506386897037955937032, 4.63666973568343889153864361843, 4.71088742390143150025999012365, 4.91370093632105664588851604055, 5.46107864327298834396471372858, 5.75759787049948060661816797671, 6.10706833429456187146494911629, 6.20595546160058381871778447327, 6.42311979002370115900379216214, 6.67850555915381894993069883876, 6.96789300546689287624223675114

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