Properties

Label 6-463e3-463.462-c0e3-0-0
Degree $6$
Conductor $99252847$
Sign $1$
Analytic cond. $0.0123371$
Root an. cond. $0.480694$
Motivic weight $0$
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 + 3·9-s − 17-s − 3·18-s + 3·25-s − 29-s − 31-s + 34-s − 43-s − 47-s + 3·49-s − 3·50-s + 58-s − 59-s − 61-s + 62-s − 67-s − 73-s − 79-s + 6·81-s + 86-s − 89-s + 94-s − 97-s − 3·98-s − 103-s − 109-s + ⋯
L(s)  = 1  − 2-s + 3·9-s − 17-s − 3·18-s + 3·25-s − 29-s − 31-s + 34-s − 43-s − 47-s + 3·49-s − 3·50-s + 58-s − 59-s − 61-s + 62-s − 67-s − 73-s − 79-s + 6·81-s + 86-s − 89-s + 94-s − 97-s − 3·98-s − 103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99252847 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99252847 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(99252847\)    =    \(463^{3}\)
Sign: $1$
Analytic conductor: \(0.0123371\)
Root analytic conductor: \(0.480694\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{463} (462, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 99252847,\ (\ :0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4005516616\)
\(L(\frac12)\) \(\approx\) \(0.4005516616\)
\(L(1)\) 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)$
bad463$C_1$ \( ( 1 - T )^{3} \)
good2$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
29$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
31$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
47$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
59$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
61$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
67$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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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

−10.02282119472721562010164826161, −9.690772800562192495320283739059, −9.420279414318351576008754865887, −9.309537406313710211994457605544, −8.802176876257533233106540220109, −8.683297336548361002958971666410, −8.564780281378534026423004403207, −7.77187975923572939972939538572, −7.67895468976345246391935296266, −7.19793993977303823030179476454, −7.07615270596553029970784112672, −6.85728469250161661112008094528, −6.54232569365252198298968946140, −6.12110202527078835275638678245, −5.45355563587983596896455434119, −5.34672852909576887496291364150, −4.52639380204327745514026020885, −4.47582534373088163512464677170, −4.45746725476707817177630453553, −3.64946712777233160145158521325, −3.42393146583910492731892658875, −2.70997577242153672759998014402, −2.19583316526044760104909394809, −1.41764096255587410441487095632, −1.32047019466465502659742302286, 1.32047019466465502659742302286, 1.41764096255587410441487095632, 2.19583316526044760104909394809, 2.70997577242153672759998014402, 3.42393146583910492731892658875, 3.64946712777233160145158521325, 4.45746725476707817177630453553, 4.47582534373088163512464677170, 4.52639380204327745514026020885, 5.34672852909576887496291364150, 5.45355563587983596896455434119, 6.12110202527078835275638678245, 6.54232569365252198298968946140, 6.85728469250161661112008094528, 7.07615270596553029970784112672, 7.19793993977303823030179476454, 7.67895468976345246391935296266, 7.77187975923572939972939538572, 8.564780281378534026423004403207, 8.683297336548361002958971666410, 8.802176876257533233106540220109, 9.309537406313710211994457605544, 9.420279414318351576008754865887, 9.690772800562192495320283739059, 10.02282119472721562010164826161

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