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Properties

Label 2-6e3-72.61-c1-0-5
Degree $2$
Conductor $216$
Sign $0.844 + 0.534i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 0.0174i)2-s + (1.99 + 0.0493i)4-s + (3.17 − 1.83i)5-s + (−0.191 + 0.332i)7-s + (−2.82 − 0.104i)8-s + (−4.51 + 2.53i)10-s + (−1.73 − 1.00i)11-s + (0.397 − 0.229i)13-s + (0.277 − 0.466i)14-s + (3.99 + 0.197i)16-s + 4.08·17-s − 4.72i·19-s + (6.43 − 3.50i)20-s + (2.44 + 1.44i)22-s + (2.97 + 5.15i)23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0123i)2-s + (0.999 + 0.0246i)4-s + (1.41 − 0.819i)5-s + (−0.0725 + 0.125i)7-s + (−0.999 − 0.0370i)8-s + (−1.42 + 0.801i)10-s + (−0.524 − 0.302i)11-s + (0.110 − 0.0636i)13-s + (0.0740 − 0.124i)14-s + (0.998 + 0.0493i)16-s + 0.990·17-s − 1.08i·19-s + (1.43 − 0.783i)20-s + (0.520 + 0.309i)22-s + (0.620 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.844 + 0.534i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.844 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937945 - 0.271894i\)
\(L(\frac12)\) \(\approx\) \(0.937945 - 0.271894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0174i)T \)
3 \( 1 \)
good5 \( 1 + (-3.17 + 1.83i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.191 - 0.332i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.397 + 0.229i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.08T + 17T^{2} \)
19 \( 1 + 4.72iT - 19T^{2} \)
23 \( 1 + (-2.97 - 5.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.03 + 1.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.592 - 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + (4.75 + 8.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.03 + 0.598i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.27 - 5.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.63iT - 53T^{2} \)
59 \( 1 + (-0.603 + 0.348i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.23 + 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.87 - 5.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (5.35 - 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 + 3.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 + (2.98 - 5.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.17993781938758679042842568219, −11.00922250567388700523031767704, −10.02500814469284266883407584059, −9.324252083071137324805260418757, −8.553946742219845490753852743883, −7.33234265746410481822121761979, −6.01798999568584744761838629784, −5.22881207841279500409641684753, −2.84905650215004957621079469695, −1.35004714164705392953881123314, 1.83115633734538758855093921274, 3.09235404359095690525709130983, 5.50152610330850081979281018156, 6.40787160577623808732622543524, 7.36937952634118316138203183104, 8.547083176876186758892577538763, 9.816815980873811105422157683310, 10.15778729078644813156666880922, 11.03064240485781031261243325912, 12.29799443032979768409020542565

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