Properties

Label 6-5070e3-1.1-c1e3-0-7
Degree $6$
Conductor $130323843000$
Sign $-1$
Analytic cond. $66352.1$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 6·4-s − 3·5-s + 9·6-s − 10·8-s + 6·9-s + 9·10-s + 2·11-s − 18·12-s + 9·15-s + 15·16-s + 3·17-s − 18·18-s − 5·19-s − 18·20-s − 6·22-s + 3·23-s + 30·24-s + 6·25-s − 10·27-s − 10·29-s − 27·30-s − 4·31-s − 21·32-s − 6·33-s − 9·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s − 3.53·8-s + 2·9-s + 2.84·10-s + 0.603·11-s − 5.19·12-s + 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s − 1.14·19-s − 4.02·20-s − 1.27·22-s + 0.625·23-s + 6.12·24-s + 6/5·25-s − 1.92·27-s − 1.85·29-s − 4.92·30-s − 0.718·31-s − 3.71·32-s − 1.04·33-s − 1.54·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(66352.1\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good7$A_4\times C_2$ \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 2 T + 18 T^{2} - 57 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 3 T + 47 T^{2} - 89 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 5 T + 63 T^{2} + 191 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 3 T + 51 T^{2} - 111 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 10 T + 104 T^{2} + 539 T^{3} + 104 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 4 T + 96 T^{2} + 247 T^{3} + 96 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 2 T + 110 T^{2} - 147 T^{3} + 110 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 17 T + 175 T^{2} + 1227 T^{3} + 175 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 19 T + 219 T^{2} - 1747 T^{3} + 219 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 7 T + 92 T^{2} + 371 T^{3} + 92 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 19 T + 221 T^{2} - 1931 T^{3} + 221 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 19 T + 253 T^{2} + 2313 T^{3} + 253 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - T + 97 T^{2} - 373 T^{3} + 97 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - T + 129 T^{2} + 35 T^{3} + 129 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - T + 99 T^{2} + 279 T^{3} + 99 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 15 T + 77 T^{2} - 47 T^{3} + 77 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 15 T + 291 T^{2} - 2383 T^{3} + 291 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 6 T + 212 T^{2} + 857 T^{3} + 212 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 9 T + 147 T^{2} + 1825 T^{3} + 147 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 4 T + 280 T^{2} - 775 T^{3} + 280 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.81595113425258159992284898349, −7.43394452507102904464981692396, −7.17244337094678118991802356400, −6.97832734829668655832231845459, −6.75446063912457778681237804636, −6.53670730220170062275208060653, −6.47304666325712688342873988536, −5.93079716826940809677992503114, −5.76777873135145552171087500729, −5.73143900944716364369844551479, −5.17818296966509723263788229667, −5.00848434070333917900898865719, −4.88072200900801362747903731677, −4.22853220196205944415798801493, −4.16667485631198866735305564024, −3.96452721617548886795754718796, −3.38199144217997126991686485417, −3.29024831008077842920425188978, −3.21734791240840005756304177954, −2.33159237780172804737933699493, −2.17147228977193801370508625623, −2.00764044666207113331524489728, −1.26360193995789821795343890824, −1.16038507217830279041073955164, −0.991237008844185505469845454745, 0, 0, 0, 0.991237008844185505469845454745, 1.16038507217830279041073955164, 1.26360193995789821795343890824, 2.00764044666207113331524489728, 2.17147228977193801370508625623, 2.33159237780172804737933699493, 3.21734791240840005756304177954, 3.29024831008077842920425188978, 3.38199144217997126991686485417, 3.96452721617548886795754718796, 4.16667485631198866735305564024, 4.22853220196205944415798801493, 4.88072200900801362747903731677, 5.00848434070333917900898865719, 5.17818296966509723263788229667, 5.73143900944716364369844551479, 5.76777873135145552171087500729, 5.93079716826940809677992503114, 6.47304666325712688342873988536, 6.53670730220170062275208060653, 6.75446063912457778681237804636, 6.97832734829668655832231845459, 7.17244337094678118991802356400, 7.43394452507102904464981692396, 7.81595113425258159992284898349

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