Properties

Label 2-6e3-1.1-c1-0-3
Degree $2$
Conductor $216$
Sign $-1$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 3·7-s − 4·11-s + 13-s + 4·17-s − 19-s − 4·23-s + 11·25-s − 4·31-s + 12·35-s − 9·37-s − 8·43-s + 12·47-s + 2·49-s + 8·53-s + 16·55-s − 4·59-s − 5·61-s − 4·65-s + 11·67-s − 8·71-s + 73-s + 12·77-s − 5·79-s − 8·83-s − 16·85-s − 12·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.13·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 11/5·25-s − 0.718·31-s + 2.02·35-s − 1.47·37-s − 1.21·43-s + 1.75·47-s + 2/7·49-s + 1.09·53-s + 2.15·55-s − 0.520·59-s − 0.640·61-s − 0.496·65-s + 1.34·67-s − 0.949·71-s + 0.117·73-s + 1.36·77-s − 0.562·79-s − 0.878·83-s − 1.73·85-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03544564786177544377995408667, −10.86882134051916494593969688660, −10.05433643313669533911783232246, −8.655801172150449552881383118942, −7.79497553082532979201942816895, −6.95550481778817521461633350929, −5.48459428248370426498378218555, −3.99250331289836498435902686175, −3.09788365205681643264055516753, 0, 3.09788365205681643264055516753, 3.99250331289836498435902686175, 5.48459428248370426498378218555, 6.95550481778817521461633350929, 7.79497553082532979201942816895, 8.655801172150449552881383118942, 10.05433643313669533911783232246, 10.86882134051916494593969688660, 12.03544564786177544377995408667

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