Properties

Label 6-825e3-1.1-c1e3-0-1
Degree $6$
Conductor $561515625$
Sign $1$
Analytic cond. $285.886$
Root an. cond. $2.56664$
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 − 3·6-s + 2·8-s + 6·9-s + 3·11-s + 2·13-s + 3·16-s + 2·17-s + 6·18-s + 8·19-s + 3·22-s − 6·24-s + 2·26-s − 10·27-s − 10·29-s + 8·31-s + 3·32-s − 9·33-s + 2·34-s + 6·37-s + 8·38-s − 6·39-s − 14·41-s − 4·43-s + 8·47-s − 9·48-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1.22·6-s + 0.707·8-s + 2·9-s + 0.904·11-s + 0.554·13-s + 3/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 0.639·22-s − 1.22·24-s + 0.392·26-s − 1.92·27-s − 1.85·29-s + 1.43·31-s + 0.530·32-s − 1.56·33-s + 0.342·34-s + 0.986·37-s + 1.29·38-s − 0.960·39-s − 2.18·41-s − 0.609·43-s + 1.16·47-s − 1.29·48-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(285.886\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.743864766\)
\(L(\frac12)\) \(\approx\) \(2.743864766\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
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

−9.509729547902967316931157586557, −8.719332441510926997684578857334, −8.409578853511103745454259503108, −8.318272012970896134030915355471, −7.80148911629081451534797706316, −7.58644271444495996613638756312, −7.26252456607141353211515782971, −6.86355015065206450571995951020, −6.65996652849161641384712338128, −6.61067391875904416570933614709, −5.82175593387592209920033989527, −5.79551129756285340068851740203, −5.66318824732485087487866037554, −5.04109463830870922650412143532, −4.88109582633404357410849875738, −4.87616480452632839857544481654, −3.93947098832744351545324993545, −3.93011457200404357013828179400, −3.90153366285414162564750922574, −3.10743837981281914408068596125, −2.86908830676011797181091771367, −1.91027492340035911089610934674, −1.67763356661225371788350724459, −1.01429119232244115145294490384, −0.72488056187630882142132887690, 0.72488056187630882142132887690, 1.01429119232244115145294490384, 1.67763356661225371788350724459, 1.91027492340035911089610934674, 2.86908830676011797181091771367, 3.10743837981281914408068596125, 3.90153366285414162564750922574, 3.93011457200404357013828179400, 3.93947098832744351545324993545, 4.87616480452632839857544481654, 4.88109582633404357410849875738, 5.04109463830870922650412143532, 5.66318824732485087487866037554, 5.79551129756285340068851740203, 5.82175593387592209920033989527, 6.61067391875904416570933614709, 6.65996652849161641384712338128, 6.86355015065206450571995951020, 7.26252456607141353211515782971, 7.58644271444495996613638756312, 7.80148911629081451534797706316, 8.318272012970896134030915355471, 8.409578853511103745454259503108, 8.719332441510926997684578857334, 9.509729547902967316931157586557

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