Properties

Label 6-57e6-1.1-c1e3-0-0
Degree $6$
Conductor $34296447249$
Sign $1$
Analytic cond. $17461.4$
Root an. cond. $5.09346$
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-s − 2·5-s − 7-s − 2·10-s − 13-s − 14-s − 16-s − 14·23-s − 3·25-s − 26-s − 4·29-s + 15·31-s + 32-s + 2·35-s − 3·37-s + 4·41-s − 3·43-s − 14·46-s + 18·47-s − 11·49-s − 3·50-s + 6·53-s − 4·58-s + 13·61-s + 15·62-s − 2·64-s + 2·65-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s − 0.377·7-s − 0.632·10-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 2.91·23-s − 3/5·25-s − 0.196·26-s − 0.742·29-s + 2.69·31-s + 0.176·32-s + 0.338·35-s − 0.493·37-s + 0.624·41-s − 0.457·43-s − 2.06·46-s + 2.62·47-s − 1.57·49-s − 0.424·50-s + 0.824·53-s − 0.525·58-s + 1.66·61-s + 1.90·62-s − 1/4·64-s + 0.248·65-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(17461.4\)
Root analytic conductor: \(5.09346\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{6} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.800786107\)
\(L(\frac12)\) \(\approx\) \(1.800786107\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
19 \( 1 \)
good2$S_4\times C_2$ \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 2 T + 7 T^{2} + 8 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + T + 12 T^{2} + 17 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + T + 18 T^{2} + 29 T^{3} + 18 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + p T^{2} )^{3} \)
23$S_4\times C_2$ \( 1 + 14 T + 97 T^{2} + 488 T^{3} + 97 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 55 T^{2} + 136 T^{3} + 55 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 15 T + 144 T^{2} - 29 p T^{3} + 144 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
41$S_4\times C_2$ \( 1 - 4 T + 91 T^{2} - 232 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 3 T + 108 T^{2} + 271 T^{3} + 108 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
53$S_4\times C_2$ \( 1 - 6 T + 87 T^{2} - 312 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 153 T^{2} - 36 T^{3} + 153 p T^{4} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 13 T + 194 T^{2} - 1513 T^{3} + 194 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 9 T + 120 T^{2} - 665 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 18 T + 225 T^{2} - 1908 T^{3} + 225 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 19 T + 302 T^{2} - 2743 T^{3} + 302 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 11 T + 240 T^{2} - 1567 T^{3} + 240 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 169 T^{2} - 472 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 16 T + 343 T^{2} - 2956 T^{3} + 343 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 2 T + 23 T^{2} - 1060 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.71435956057497221456898792166, −7.58001348248452133358347317486, −7.01428097868081438510683021992, −6.84342452620837481203659925822, −6.71592137314820929266741566390, −6.28262575387051116509695506992, −6.24442470622530667956531931773, −5.79815328535240558271846677929, −5.65370879349837794886391780141, −5.39826874295739582424784674223, −4.87562099503737659412609585312, −4.78693825267277723977174248434, −4.61771212231854216260714124870, −3.94720159198705705690894644285, −3.92867131592538724401952333760, −3.92510201337969948468173452061, −3.56771318262696213402071862467, −3.21822089476621381776739753226, −2.58908043888323618132161169445, −2.44929385036094861160676087479, −2.16246149303960155526409308322, −1.95717406589227275335648745645, −1.15722349165664516667868881151, −0.78156480231356303898854791481, −0.29627147163659246408247627009, 0.29627147163659246408247627009, 0.78156480231356303898854791481, 1.15722349165664516667868881151, 1.95717406589227275335648745645, 2.16246149303960155526409308322, 2.44929385036094861160676087479, 2.58908043888323618132161169445, 3.21822089476621381776739753226, 3.56771318262696213402071862467, 3.92510201337969948468173452061, 3.92867131592538724401952333760, 3.94720159198705705690894644285, 4.61771212231854216260714124870, 4.78693825267277723977174248434, 4.87562099503737659412609585312, 5.39826874295739582424784674223, 5.65370879349837794886391780141, 5.79815328535240558271846677929, 6.24442470622530667956531931773, 6.28262575387051116509695506992, 6.71592137314820929266741566390, 6.84342452620837481203659925822, 7.01428097868081438510683021992, 7.58001348248452133358347317486, 7.71435956057497221456898792166

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