Properties

Label 6-507e3-1.1-c1e3-0-1
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  + 2-s + 3·3-s − 3·4-s + 4·5-s + 3·6-s + 10·7-s − 4·8-s + 6·9-s + 4·10-s − 11-s − 9·12-s + 10·14-s + 12·15-s + 3·16-s − 7·17-s + 6·18-s + 11·19-s − 12·20-s + 30·21-s − 22-s + 2·23-s − 12·24-s − 2·25-s + 10·27-s − 30·28-s − 8·29-s + 12·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 3/2·4-s + 1.78·5-s + 1.22·6-s + 3.77·7-s − 1.41·8-s + 2·9-s + 1.26·10-s − 0.301·11-s − 2.59·12-s + 2.67·14-s + 3.09·15-s + 3/4·16-s − 1.69·17-s + 1.41·18-s + 2.52·19-s − 2.68·20-s + 6.54·21-s − 0.213·22-s + 0.417·23-s − 2.44·24-s − 2/5·25-s + 1.92·27-s − 5.66·28-s − 1.48·29-s + 2.19·30-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

\(L(1)\) \(\approx\) \(9.085268571\)
\(L(\frac12)\) \(\approx\) \(9.085268571\)
\(L(\frac{3}{2})\) 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$A_4\times C_2$ \( 1 - T + p^{2} T^{2} - 3 T^{3} + p^{3} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 - 4 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - 10 T + 52 T^{2} - 169 T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + T + 3 T^{2} - 21 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 11 T + 67 T^{2} - 305 T^{3} + 67 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 2 T + 26 T^{2} - 175 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 8 T + 70 T^{2} - 299 T^{3} + 70 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 14 T + 174 T^{2} - 1127 T^{3} + 174 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - T + 121 T^{2} - 81 T^{3} + 121 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 3 T + 104 T^{2} + 287 T^{3} + 104 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 9 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 14 T + 3 p T^{2} + 1596 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 5 T + 179 T^{2} - 573 T^{3} + 179 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 6 T + 134 T^{2} + 391 T^{3} + 134 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 18 T + 320 T^{2} - 2795 T^{3} + 320 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 16 T + 304 T^{2} + 2613 T^{3} + 304 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 5 T + 259 T^{2} + 891 T^{3} + 259 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 5 T + 122 T^{2} + 219 T^{3} + 122 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.561021900356378255153181199683, −9.262435606503044850961956068134, −9.155827761908500415080014517113, −9.100260176152997461158262385354, −8.366793768393057078528770929787, −8.203020580658235864380668168826, −8.025260521216847421627443907545, −7.87557052902609569213810450803, −7.39260193239174232205051567090, −7.35323204584666956427716043452, −6.42786441842051864421539082363, −6.25426652103977034272802957201, −5.71329151041676788344608061322, −5.29296223508686036055147160322, −5.09179957740881804876112480392, −4.97181311706309888084725703013, −4.40429600862828136221886910326, −4.28595217044612382967827364550, −4.17826828325452043239009646316, −3.20298903581918346857264570917, −3.01693015539831466587929705758, −2.31169801433556225197642981285, −1.98368280574573453517805604990, −1.48266702256645694994149951192, −1.36618448393371556817901132706, 1.36618448393371556817901132706, 1.48266702256645694994149951192, 1.98368280574573453517805604990, 2.31169801433556225197642981285, 3.01693015539831466587929705758, 3.20298903581918346857264570917, 4.17826828325452043239009646316, 4.28595217044612382967827364550, 4.40429600862828136221886910326, 4.97181311706309888084725703013, 5.09179957740881804876112480392, 5.29296223508686036055147160322, 5.71329151041676788344608061322, 6.25426652103977034272802957201, 6.42786441842051864421539082363, 7.35323204584666956427716043452, 7.39260193239174232205051567090, 7.87557052902609569213810450803, 8.025260521216847421627443907545, 8.203020580658235864380668168826, 8.366793768393057078528770929787, 9.100260176152997461158262385354, 9.155827761908500415080014517113, 9.262435606503044850961956068134, 9.561021900356378255153181199683

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