Properties

Label 6-19e6-1.1-c1e3-0-0
Degree $6$
Conductor $47045881$
Sign $1$
Analytic cond. $23.9526$
Root an. cond. $1.69782$
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  + 3·2-s + 3·3-s + 3·4-s − 3·5-s + 9·6-s − 9·10-s + 9·12-s − 9·15-s − 3·16-s + 6·17-s − 9·20-s − 6·23-s − 6·25-s − 10·27-s + 15·29-s − 27·30-s + 9·31-s − 6·32-s + 18·34-s + 12·41-s − 18·46-s − 6·47-s − 9·48-s − 18·49-s − 18·50-s + 18·51-s + 6·53-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s + 3.67·6-s − 2.84·10-s + 2.59·12-s − 2.32·15-s − 3/4·16-s + 1.45·17-s − 2.01·20-s − 1.25·23-s − 6/5·25-s − 1.92·27-s + 2.78·29-s − 4.92·30-s + 1.61·31-s − 1.06·32-s + 3.08·34-s + 1.87·41-s − 2.65·46-s − 0.875·47-s − 1.29·48-s − 2.57·49-s − 2.54·50-s + 2.52·51-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(47045881\)    =    \(19^{6}\)
Sign: $1$
Analytic conductor: \(23.9526\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{361} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 47045881,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.850860565\)
\(L(\frac12)\) \(\approx\) \(5.850860565\)
\(L(\frac{3}{2})\) 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)$
bad19 \( 1 \)
good2$A_4\times C_2$ \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
3$A_4\times C_2$ \( 1 - p T + p^{2} T^{2} - 17 T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 18 T^{2} + T^{3} + 18 p T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 + 18 T^{2} - 37 T^{3} + 18 p T^{4} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 252 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 15 T + 159 T^{2} - 981 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 9 T + 99 T^{2} - 505 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 90 T^{2} + 17 T^{3} + 90 p T^{4} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 12 T + 132 T^{2} - 873 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 72 T^{2} + 163 T^{3} + 72 p T^{4} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 6 T + 132 T^{2} + 567 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 6 T + 150 T^{2} - 585 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 21 T + 312 T^{2} - 2745 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 9 T + 162 T^{2} - 917 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 18 T + 225 T^{2} + 1988 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 30 T + 501 T^{2} - 5148 T^{3} + 501 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 171 T^{2} + 64 T^{3} + 171 p T^{4} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 9 T + 135 T^{2} - 613 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 60 T^{2} + 459 T^{3} + 60 p T^{4} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 15 T + 321 T^{2} - 2727 T^{3} + 321 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 15 T + 330 T^{2} + 2783 T^{3} + 330 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.13895912933182563538853339236, −9.826649694326291032326070763208, −9.621167918096850934933523719004, −9.373739471293226371191488784257, −8.688751395806160089651522372028, −8.532441758409032066690774716175, −8.165051593230146577947045230875, −7.960314767871374497266413243405, −7.925179242709707989742441980582, −7.63959083352209911361478424563, −6.75343168049276423687964865029, −6.61698226712208514730901470996, −6.28526575603015736189868847262, −5.68452963345921361224815461830, −5.34515168780847009755595846648, −5.26288813311469045183077624962, −4.43041809116077034329245292354, −4.27223811406285223842955492809, −4.20437971502618059891215490589, −3.42080236544457324465420890675, −3.35345298770707616896232966929, −3.15907241361875228465613894042, −2.33778935727498860662206296797, −2.27155535338614360775181545455, −0.815433306080246256221657145441, 0.815433306080246256221657145441, 2.27155535338614360775181545455, 2.33778935727498860662206296797, 3.15907241361875228465613894042, 3.35345298770707616896232966929, 3.42080236544457324465420890675, 4.20437971502618059891215490589, 4.27223811406285223842955492809, 4.43041809116077034329245292354, 5.26288813311469045183077624962, 5.34515168780847009755595846648, 5.68452963345921361224815461830, 6.28526575603015736189868847262, 6.61698226712208514730901470996, 6.75343168049276423687964865029, 7.63959083352209911361478424563, 7.925179242709707989742441980582, 7.960314767871374497266413243405, 8.165051593230146577947045230875, 8.532441758409032066690774716175, 8.688751395806160089651522372028, 9.373739471293226371191488784257, 9.621167918096850934933523719004, 9.826649694326291032326070763208, 10.13895912933182563538853339236

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