Properties

Label 6-8550e3-1.1-c1e3-0-8
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
Motivic weight $1$
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 + 6·4-s + 10·8-s + 4·11-s − 6·13-s + 15·16-s + 10·17-s − 3·19-s + 12·22-s + 6·23-s − 18·26-s − 10·29-s + 14·31-s + 21·32-s + 30·34-s − 2·37-s − 9·38-s + 10·43-s + 24·44-s + 18·46-s − 2·47-s − 5·49-s − 36·52-s + 14·53-s − 30·58-s − 14·59-s + 6·61-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 3.53·8-s + 1.20·11-s − 1.66·13-s + 15/4·16-s + 2.42·17-s − 0.688·19-s + 2.55·22-s + 1.25·23-s − 3.53·26-s − 1.85·29-s + 2.51·31-s + 3.71·32-s + 5.14·34-s − 0.328·37-s − 1.45·38-s + 1.52·43-s + 3.61·44-s + 2.65·46-s − 0.291·47-s − 5/7·49-s − 4.99·52-s + 1.92·53-s − 3.93·58-s − 1.82·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(38.11225501\)
\(L(\frac12)\) \(\approx\) \(38.11225501\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 10 T + 71 T^{2} - 332 T^{3} + 71 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 6 T + 65 T^{2} - 268 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 121 T^{2} - 716 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 2 T + 27 T^{2} + 380 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 107 T^{2} + 16 T^{3} + 107 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 125 T^{2} - 852 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 2 T + 89 T^{2} + 4 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 171 T^{2} - 1332 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 14 T + 229 T^{2} + 1692 T^{3} + 229 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 133 T^{2} - 504 T^{3} + 133 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 8 T + 155 T^{2} - 912 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 22 T + 361 T^{2} - 3676 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 129 T^{2} + 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 4 T + 219 T^{2} + 792 T^{3} + 219 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 111 T^{2} - 948 T^{3} + 111 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.93139190892876434833633780553, −6.43770541911062514724179094641, −6.35158800122379895236826127811, −6.20278028738625495626483851238, −5.81227021516801464019663748728, −5.67095066648389555463350283542, −5.46035528237938475064607924799, −5.18694091754342335729730917047, −4.91154918109079657644833911619, −4.79931281760801231230335785800, −4.48928668182503240168271057512, −4.43679769515105157634809179422, −3.89250180232438592115704161965, −3.68454686611964983526634557617, −3.63741852785171153819467042593, −3.37809071150711237115101466420, −2.89515991568719510181861576381, −2.79142223245457426799506676119, −2.64614969208033695618577754784, −2.03733770360757473493935637481, −1.89404072691630753816024421186, −1.78898808404370800206245016115, −1.05221846511143965462223137256, −0.75417171555968056164327999379, −0.69543337658002509181844921460, 0.69543337658002509181844921460, 0.75417171555968056164327999379, 1.05221846511143965462223137256, 1.78898808404370800206245016115, 1.89404072691630753816024421186, 2.03733770360757473493935637481, 2.64614969208033695618577754784, 2.79142223245457426799506676119, 2.89515991568719510181861576381, 3.37809071150711237115101466420, 3.63741852785171153819467042593, 3.68454686611964983526634557617, 3.89250180232438592115704161965, 4.43679769515105157634809179422, 4.48928668182503240168271057512, 4.79931281760801231230335785800, 4.91154918109079657644833911619, 5.18694091754342335729730917047, 5.46035528237938475064607924799, 5.67095066648389555463350283542, 5.81227021516801464019663748728, 6.20278028738625495626483851238, 6.35158800122379895236826127811, 6.43770541911062514724179094641, 6.93139190892876434833633780553

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