Properties

Label 6-1215e3-15.14-c0e3-0-0
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

Downloads

Learn more

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

\(L(\frac{1}{2})\) \(\approx\) \(0.6607072100\)
\(L(\frac12)\) \(\approx\) \(0.6607072100\)
\(L(1)\) 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$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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

−8.848975733966765300053879982991, −8.621696821443403005276224123950, −8.291867759144774257263739097355, −7.81414878883878292461954496392, −7.80423598441035057704704438528, −7.41635556506473830273969777476, −7.33006665411073151930560875668, −7.03114400708906439862728711072, −7.00484362942174626563346358042, −6.36149557355958041185135603413, −6.08287050773859682180577955196, −5.57149517508684153995894669899, −5.52228458483151498142393868828, −4.70103732842276672144431658969, −4.69000145792961307767509751242, −4.62422241423623700054177698200, −3.94716475438909020550293766369, −3.92297665967567896912010639822, −3.69978449927477500861679657673, −3.19662269227392008259292455690, −2.85880377574486163417854967499, −2.38671046857683219749119513085, −1.98247246182906572586659465493, −0.938118664095298493348676384056, −0.895066291246857968919851878791, 0.895066291246857968919851878791, 0.938118664095298493348676384056, 1.98247246182906572586659465493, 2.38671046857683219749119513085, 2.85880377574486163417854967499, 3.19662269227392008259292455690, 3.69978449927477500861679657673, 3.92297665967567896912010639822, 3.94716475438909020550293766369, 4.62422241423623700054177698200, 4.69000145792961307767509751242, 4.70103732842276672144431658969, 5.52228458483151498142393868828, 5.57149517508684153995894669899, 6.08287050773859682180577955196, 6.36149557355958041185135603413, 7.00484362942174626563346358042, 7.03114400708906439862728711072, 7.33006665411073151930560875668, 7.41635556506473830273969777476, 7.80423598441035057704704438528, 7.81414878883878292461954496392, 8.291867759144774257263739097355, 8.621696821443403005276224123950, 8.848975733966765300053879982991

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