Properties

Label 6-1815e3-1.1-c1e3-0-2
Degree $6$
Conductor $5979018375$
Sign $1$
Analytic cond. $3044.11$
Root an. cond. $3.80694$
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  + 2-s + 3·3-s + 3·5-s + 3·6-s + 2·8-s + 6·9-s + 3·10-s + 2·13-s + 9·15-s + 3·16-s + 2·17-s + 6·18-s − 8·19-s + 6·24-s + 6·25-s + 2·26-s + 10·27-s + 10·29-s + 9·30-s + 8·31-s + 3·32-s + 2·34-s − 6·37-s − 8·38-s + 6·39-s + 6·40-s + 14·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.34·5-s + 1.22·6-s + 0.707·8-s + 2·9-s + 0.948·10-s + 0.554·13-s + 2.32·15-s + 3/4·16-s + 0.485·17-s + 1.41·18-s − 1.83·19-s + 1.22·24-s + 6/5·25-s + 0.392·26-s + 1.92·27-s + 1.85·29-s + 1.64·30-s + 1.43·31-s + 0.530·32-s + 0.342·34-s − 0.986·37-s − 1.29·38-s + 0.960·39-s + 0.948·40-s + 2.18·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(3044.11\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.03420967\)
\(L(\frac12)\) \(\approx\) \(18.03420967\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good2$S_4\times C_2$ \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T - T^{2} + 116 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 8 T + 41 T^{2} + 144 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 5 T^{2} - 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
41$S_4\times C_2$ \( 1 - 14 T + 167 T^{2} - 1156 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 49 T^{2} - 56 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 107 T^{2} + 644 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 4 T + 153 T^{2} + 472 T^{3} + 153 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 14 T + 223 T^{2} - 1700 T^{3} + 223 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 173 T^{2} + 1096 T^{3} + 173 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 10 T + 215 T^{2} + 1580 T^{3} + 215 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 399 T^{2} - 4276 T^{3} + 399 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.417074726670171228732884791717, −8.064980757721626144435657038888, −7.87032407848544047831295562515, −7.30969757836230906365548789429, −7.27459501172959312068476464470, −6.88161925653178687353406807453, −6.40870015602004296514712804773, −6.37803453785360511975715515429, −6.18784451316645286782615283200, −5.96303005005814666030707535135, −5.21771802521927821732912709099, −5.04916161274286318029216841143, −5.01621108430893583738914439387, −4.49712345373581957621470083778, −4.17949852822205665608011889483, −4.14849204192772368969300782728, −3.61665316948219825277698527930, −3.33565697388660835294905882749, −2.95190567342759080754221366648, −2.54783851640045276144847827202, −2.49282677479250398565505451754, −1.84048687588972411031430750988, −1.81019927113047872090365336656, −1.14030064010705934337546709362, −0.832755459395975989292325862828, 0.832755459395975989292325862828, 1.14030064010705934337546709362, 1.81019927113047872090365336656, 1.84048687588972411031430750988, 2.49282677479250398565505451754, 2.54783851640045276144847827202, 2.95190567342759080754221366648, 3.33565697388660835294905882749, 3.61665316948219825277698527930, 4.14849204192772368969300782728, 4.17949852822205665608011889483, 4.49712345373581957621470083778, 5.01621108430893583738914439387, 5.04916161274286318029216841143, 5.21771802521927821732912709099, 5.96303005005814666030707535135, 6.18784451316645286782615283200, 6.37803453785360511975715515429, 6.40870015602004296514712804773, 6.88161925653178687353406807453, 7.27459501172959312068476464470, 7.30969757836230906365548789429, 7.87032407848544047831295562515, 8.064980757721626144435657038888, 8.417074726670171228732884791717

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