Properties

Label 6-507e3-1.1-c1e3-0-0
Degree $6$
Conductor $130323843$
Sign $1$
Analytic cond. $66.3521$
Root an. cond. $2.01206$
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 + 7·4-s + 6·5-s − 9·6-s + 2·7-s + 14·8-s + 6·9-s + 18·10-s + 5·11-s − 21·12-s + 6·14-s − 18·15-s + 21·16-s − 17-s + 18·18-s − 7·19-s + 42·20-s − 6·21-s + 15·22-s − 42·24-s + 16·25-s − 10·27-s + 14·28-s − 2·29-s − 54·30-s − 16·31-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 7/2·4-s + 2.68·5-s − 3.67·6-s + 0.755·7-s + 4.94·8-s + 2·9-s + 5.69·10-s + 1.50·11-s − 6.06·12-s + 1.60·14-s − 4.64·15-s + 21/4·16-s − 0.242·17-s + 4.24·18-s − 1.60·19-s + 9.39·20-s − 1.30·21-s + 3.19·22-s − 8.57·24-s + 16/5·25-s − 1.92·27-s + 2.64·28-s − 0.371·29-s − 9.85·30-s − 2.87·31-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(66.3521\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.26539294\)
\(L(\frac12)\) \(\approx\) \(11.26539294\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good2$C_6$ \( 1 - 3 T + p T^{2} + T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 - 6 T + 4 p T^{2} - 47 T^{3} + 4 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - 2 T + 20 T^{2} - 27 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 5 T + 25 T^{2} - 69 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 7 T + 71 T^{2} + 273 T^{3} + 71 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 20 T^{2} + 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 16 T + 134 T^{2} + 795 T^{3} + 134 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 22 T + 270 T^{2} - 2005 T^{3} + 270 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 11 T + 147 T^{2} - 873 T^{3} + 147 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 15 T + 176 T^{2} + 1331 T^{3} + 176 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 7 T + 155 T^{2} - 665 T^{3} + 155 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 6 T + 161 T^{2} + 604 T^{3} + 161 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 11 T + 155 T^{2} - 1515 T^{3} + 155 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 122 T^{2} + 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 6 T + 84 T^{2} + 47 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 12 T + 290 T^{2} - 2035 T^{3} + 290 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - T + 167 T^{2} - 65 T^{3} + 167 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 5 T + 10 T^{2} - 667 T^{3} + 10 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.718590993472648888920182265333, −9.541523968179841066512171474452, −9.365669293112238223844295430788, −9.207476267830635728608549577461, −8.506509273980416654680785872432, −7.931350372598345093761783796969, −7.69722221732747243271926069083, −7.34056233531221372370716228875, −6.82303400118216542485672566361, −6.49836467154880290748927830507, −6.45385532909737390433362921761, −6.13159882277292972167793745591, −6.08007829497662988317020565119, −5.58778236919420814783250669715, −5.34355285688175209912160995369, −5.06275700736903279327421974501, −4.47555183432868305616718971206, −4.31631134428115154031449675131, −4.29783117408011316016719717866, −3.41217922415537127594478474991, −3.00726401005179861311751062407, −2.05843232776131367445206750053, −1.96556914197670966460621889314, −1.87187510316131161114710467104, −1.25195005858148234988264507219, 1.25195005858148234988264507219, 1.87187510316131161114710467104, 1.96556914197670966460621889314, 2.05843232776131367445206750053, 3.00726401005179861311751062407, 3.41217922415537127594478474991, 4.29783117408011316016719717866, 4.31631134428115154031449675131, 4.47555183432868305616718971206, 5.06275700736903279327421974501, 5.34355285688175209912160995369, 5.58778236919420814783250669715, 6.08007829497662988317020565119, 6.13159882277292972167793745591, 6.45385532909737390433362921761, 6.49836467154880290748927830507, 6.82303400118216542485672566361, 7.34056233531221372370716228875, 7.69722221732747243271926069083, 7.931350372598345093761783796969, 8.506509273980416654680785872432, 9.207476267830635728608549577461, 9.365669293112238223844295430788, 9.541523968179841066512171474452, 9.718590993472648888920182265333

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