Properties

Label 6-8550e3-1.1-c1e3-0-17
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s + 10·8-s − 8·11-s − 4·13-s + 15·16-s + 2·17-s + 3·19-s − 24·22-s − 8·23-s − 12·26-s − 10·29-s + 21·32-s + 6·34-s − 4·37-s + 9·38-s − 4·41-s + 6·43-s − 48·44-s − 24·46-s + 8·47-s − 5·49-s − 24·52-s − 18·53-s − 30·58-s − 10·59-s + 14·61-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 3.53·8-s − 2.41·11-s − 1.10·13-s + 15/4·16-s + 0.485·17-s + 0.688·19-s − 5.11·22-s − 1.66·23-s − 2.35·26-s − 1.85·29-s + 3.71·32-s + 1.02·34-s − 0.657·37-s + 1.45·38-s − 0.624·41-s + 0.914·43-s − 7.23·44-s − 3.53·46-s + 1.16·47-s − 5/7·49-s − 3.32·52-s − 2.47·53-s − 3.93·58-s − 1.30·59-s + 1.79·61-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: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 + 8 T + 41 T^{2} + 160 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 23 T^{2} + 72 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 31 T^{2} - 60 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 8 T + 77 T^{2} + 352 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 107 T^{2} + 572 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 43 T^{2} - 72 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 6 T + 5 T^{2} + 244 T^{3} + 5 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 149 T^{2} - 736 T^{3} + 149 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 + 10 T + 173 T^{2} + 1172 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 20 T + 249 T^{2} + 2360 T^{3} + 249 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 20 T + 261 T^{2} + 2520 T^{3} + 261 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 + 221 T^{2} - 16 T^{3} + 221 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 73 T^{2} + 504 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 24 T + 443 T^{2} + 4672 T^{3} + 443 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 22 T + 439 T^{2} + 4564 T^{3} + 439 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.26797251122725166331103125536, −6.89822281176198746450474823625, −6.68659734201468240565283234543, −6.58426748182785829032276164509, −5.97143893759127036427898969016, −5.92191343565392045173259298155, −5.68941102478384567301592458254, −5.50325262314081907050185871832, −5.37587375737510482844118757425, −5.33616296523912045398659182247, −4.76463192728102648060899019265, −4.59003287182386651509907057275, −4.50339253326876643345873624053, −4.22268787188847659401012501087, −3.89957349546523834065971345229, −3.71752959598694663524365417191, −3.20579710747850825251153755612, −3.08883234970793877900312336878, −2.98159096782937508208356549140, −2.53408084748291412134699387442, −2.43536516966253238008759609258, −2.21215486267884727371759061022, −1.63857972936392363372430223311, −1.40302578587361448654865435428, −1.34820802244291330751383555986, 0, 0, 0, 1.34820802244291330751383555986, 1.40302578587361448654865435428, 1.63857972936392363372430223311, 2.21215486267884727371759061022, 2.43536516966253238008759609258, 2.53408084748291412134699387442, 2.98159096782937508208356549140, 3.08883234970793877900312336878, 3.20579710747850825251153755612, 3.71752959598694663524365417191, 3.89957349546523834065971345229, 4.22268787188847659401012501087, 4.50339253326876643345873624053, 4.59003287182386651509907057275, 4.76463192728102648060899019265, 5.33616296523912045398659182247, 5.37587375737510482844118757425, 5.50325262314081907050185871832, 5.68941102478384567301592458254, 5.92191343565392045173259298155, 5.97143893759127036427898969016, 6.58426748182785829032276164509, 6.68659734201468240565283234543, 6.89822281176198746450474823625, 7.26797251122725166331103125536

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