Properties

Label 6-825e3-1.1-c1e3-0-5
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 2·4-s + 3·6-s − 4·7-s + 2·8-s + 6·9-s + 3·11-s + 6·12-s − 2·13-s + 4·14-s + 16-s − 6·18-s − 6·19-s + 12·21-s − 3·22-s − 12·23-s − 6·24-s + 2·26-s − 10·27-s + 8·28-s + 8·29-s − 8·31-s + 32-s − 9·33-s − 12·36-s − 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 1.51·7-s + 0.707·8-s + 2·9-s + 0.904·11-s + 1.73·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.41·18-s − 1.37·19-s + 2.61·21-s − 0.639·22-s − 2.50·23-s − 1.22·24-s + 0.392·26-s − 1.92·27-s + 1.51·28-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 1.56·33-s − 2·36-s − 0.657·37-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: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 4 T + 3 p T^{2} + 52 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 35 T^{2} + 48 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 23 T^{2} - 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 - 8 T + 71 T^{2} - 304 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 - 8 T + 11 T^{2} + 272 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 884 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 16 T + 189 T^{2} + 1536 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 16 T + 191 T^{2} + 1712 T^{3} + 191 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 22 T + 291 T^{2} + 2692 T^{3} + 291 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T + 185 T^{2} + 1544 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 117 T^{2} + 760 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 175 T^{2} - 1072 T^{3} + 175 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 10 T + 25 T^{2} - 140 T^{3} + 25 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 189 T^{2} - 1056 T^{3} + 189 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 255 T^{2} - 2684 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 16 T + 131 T^{2} + 672 T^{3} + 131 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.667750402935049765515411567581, −9.255685963136796812018701128104, −9.083657907431766356766631054875, −8.958755080134150296825081630491, −8.214905430638894958424429450593, −8.152398093026085170495732770405, −8.068013884955445269719119145717, −7.39033607072128127597643024710, −7.27222957457784295880375963242, −6.66540924706850267965456337882, −6.41431558191273704550629764140, −6.28561739412942160519080176542, −6.28062773744403945913256246318, −5.88651321407098908231315876578, −5.24767170607294390761269817532, −5.09996759719112183530139638036, −4.64017506545693089220384394903, −4.38568330778879800444946357376, −4.24391503015009162106945243657, −3.61582496401322644909572348923, −3.41778385349178823269438616811, −2.97910414254756085509959498399, −2.22506912740385525617645534196, −1.63690961981755026140006473248, −1.36951501118679666259017721234, 0, 0, 0, 1.36951501118679666259017721234, 1.63690961981755026140006473248, 2.22506912740385525617645534196, 2.97910414254756085509959498399, 3.41778385349178823269438616811, 3.61582496401322644909572348923, 4.24391503015009162106945243657, 4.38568330778879800444946357376, 4.64017506545693089220384394903, 5.09996759719112183530139638036, 5.24767170607294390761269817532, 5.88651321407098908231315876578, 6.28062773744403945913256246318, 6.28561739412942160519080176542, 6.41431558191273704550629764140, 6.66540924706850267965456337882, 7.27222957457784295880375963242, 7.39033607072128127597643024710, 8.068013884955445269719119145717, 8.152398093026085170495732770405, 8.214905430638894958424429450593, 8.958755080134150296825081630491, 9.083657907431766356766631054875, 9.255685963136796812018701128104, 9.667750402935049765515411567581

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