Properties

Label 6-1215e3-15.14-c0e3-0-1
Degree $6$
Conductor $1793613375$
Sign $1$
Analytic cond. $0.222946$
Root an. cond. $0.778693$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 8-s + 6·25-s − 3·40-s − 3·47-s + 3·49-s − 3·107-s − 3·113-s + 3·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 3·5-s − 8-s + 6·25-s − 3·40-s − 3·47-s + 3·49-s − 3·107-s − 3·113-s + 3·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{15} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(0.222946\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1215} (1214, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{15} \cdot 5^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.771257637\)
\(L(\frac12)\) \(\approx\) \(1.771257637\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
good2$C_6$ \( 1 + T^{3} + T^{6} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
17$C_6$ \( 1 + T^{3} + T^{6} \)
19$C_6$ \( 1 + T^{3} + T^{6} \)
23$C_6$ \( 1 + T^{3} + T^{6} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_6$ \( 1 + T^{3} + T^{6} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
47$C_2$ \( ( 1 + T + T^{2} )^{3} \)
53$C_6$ \( 1 + T^{3} + T^{6} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_6$ \( 1 + T^{3} + T^{6} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
79$C_6$ \( 1 + T^{3} + T^{6} \)
83$C_6$ \( 1 + T^{3} + T^{6} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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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

−9.041142430985860727971389945400, −8.610667495590115767976512760706, −8.440280289745963349038572417020, −8.298220456246289278574709478254, −7.61319724634937674959421002224, −7.52402586814344603989073733040, −6.97416443656283533655506106178, −6.75516531538851974719475848834, −6.54746387037460936748054246189, −6.29199373399859431863493072110, −6.05535709546618264787834007193, −5.64159988645922915140656906872, −5.60670509948447307850626711930, −5.19440990583355660462947601039, −4.99674377321321699744893512959, −4.69352110432006148169491239553, −4.13289750210627650038671715678, −3.81434641916160081822569221868, −3.25918029419843707722458492791, −2.89755232741779050442988873666, −2.73111280456924107711235754465, −2.21724250953410198289258372309, −2.07239124287941235897836437531, −1.36087344442962051706943086223, −1.19443623302098556629034947050, 1.19443623302098556629034947050, 1.36087344442962051706943086223, 2.07239124287941235897836437531, 2.21724250953410198289258372309, 2.73111280456924107711235754465, 2.89755232741779050442988873666, 3.25918029419843707722458492791, 3.81434641916160081822569221868, 4.13289750210627650038671715678, 4.69352110432006148169491239553, 4.99674377321321699744893512959, 5.19440990583355660462947601039, 5.60670509948447307850626711930, 5.64159988645922915140656906872, 6.05535709546618264787834007193, 6.29199373399859431863493072110, 6.54746387037460936748054246189, 6.75516531538851974719475848834, 6.97416443656283533655506106178, 7.52402586814344603989073733040, 7.61319724634937674959421002224, 8.298220456246289278574709478254, 8.440280289745963349038572417020, 8.610667495590115767976512760706, 9.041142430985860727971389945400

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