Properties

Label 6-536e3-536.133-c0e3-0-0
Degree $6$
Conductor $153990656$
Sign $1$
Analytic cond. $0.0191410$
Root an. cond. $0.517202$
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  − 3·2-s + 3-s + 6·4-s + 5-s − 3·6-s − 10·8-s − 3·10-s + 11-s + 6·12-s + 13-s + 15-s + 15·16-s − 17-s + 6·20-s − 3·22-s − 23-s − 10·24-s − 3·26-s − 3·30-s − 21·32-s + 33-s + 3·34-s + 39-s − 10·40-s + 43-s + 6·44-s + 3·46-s + ⋯
L(s)  = 1  − 3·2-s + 3-s + 6·4-s + 5-s − 3·6-s − 10·8-s − 3·10-s + 11-s + 6·12-s + 13-s + 15-s + 15·16-s − 17-s + 6·20-s − 3·22-s − 23-s − 10·24-s − 3·26-s − 3·30-s − 21·32-s + 33-s + 3·34-s + 39-s − 10·40-s + 43-s + 6·44-s + 3·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 67^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 67^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 67^{3}\)
Sign: $1$
Analytic conductor: \(0.0191410\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{536} (133, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 67^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3306896563\)
\(L(\frac12)\) \(\approx\) \(0.3306896563\)
\(L(1)\) 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)$
bad2$C_1$ \( ( 1 + T )^{3} \)
67$C_1$ \( ( 1 + T )^{3} \)
good3$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
5$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
13$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
17$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
23$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
47$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
53$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
71$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
73$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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

−10.01004310592409514211731123294, −9.254848184169428343740878709725, −9.134732906217286538180687485268, −8.981041046042858229249609130945, −8.941668702027069244549406045497, −8.460983270798979522927504731056, −8.387050183722462468831537672236, −7.80772800895855874971029050887, −7.73067948880539300272265776007, −7.16143587950239904710736941497, −7.09852289495463052425338075446, −6.69541207727282465800285058801, −6.29751677501735064867847053492, −5.96562901123342482248701724320, −5.85747932399892133889911789951, −5.63226423934626143401753205157, −4.81108574171467223934865707331, −3.99809265331282871294376617322, −3.83257673977440739533329411334, −3.32176610777350327064755448938, −2.62428330104468251834559643497, −2.53382066237305735866409952275, −2.18569641276333263378569885026, −1.37045074564157809475076615507, −1.36077569376897373138759855754, 1.36077569376897373138759855754, 1.37045074564157809475076615507, 2.18569641276333263378569885026, 2.53382066237305735866409952275, 2.62428330104468251834559643497, 3.32176610777350327064755448938, 3.83257673977440739533329411334, 3.99809265331282871294376617322, 4.81108574171467223934865707331, 5.63226423934626143401753205157, 5.85747932399892133889911789951, 5.96562901123342482248701724320, 6.29751677501735064867847053492, 6.69541207727282465800285058801, 7.09852289495463052425338075446, 7.16143587950239904710736941497, 7.73067948880539300272265776007, 7.80772800895855874971029050887, 8.387050183722462468831537672236, 8.460983270798979522927504731056, 8.941668702027069244549406045497, 8.981041046042858229249609130945, 9.134732906217286538180687485268, 9.254848184169428343740878709725, 10.01004310592409514211731123294

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