Properties

Label 2-3e3-1.1-c1-0-0
Degree $2$
Conductor $27$
Sign $1$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s + 2·28-s − 4·31-s + 11·37-s + 8·43-s − 6·49-s − 10·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 14·76-s + 17·79-s − 5·91-s − 19·97-s + 10·100-s − 13·103-s + 2·109-s − 4·112-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s + 0.377·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 1.60·76-s + 1.91·79-s − 0.524·91-s − 1.92·97-s + 100-s − 1.28·103-s + 0.191·109-s − 0.377·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{27} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5888795834\)
\(L(\frac12)\) \(\approx\) \(0.5888795834\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−17.54958210256512249523818407718, −16.30001713525188163931408346208, −14.89211076740924954992271086519, −13.58211369833951574831601484907, −12.71563949069013251763955822657, −10.90872829298908742564232271224, −9.429199208210393651255146412199, −8.217650367462526737991465554229, −6.04893540000987436402649531985, −4.04304401379743272242521882180, 4.04304401379743272242521882180, 6.04893540000987436402649531985, 8.217650367462526737991465554229, 9.429199208210393651255146412199, 10.90872829298908742564232271224, 12.71563949069013251763955822657, 13.58211369833951574831601484907, 14.89211076740924954992271086519, 16.30001713525188163931408346208, 17.54958210256512249523818407718

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