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Properties

Label 2-12e3-8.3-c0-0-0
Degree $2$
Conductor $1728$
Sign $-0.707 - 0.707i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·5-s + i·7-s − 1.73·11-s − 1.99·25-s − i·31-s − 1.73·35-s + 1.73i·53-s − 2.99i·55-s + 73-s − 1.73i·77-s + 2i·79-s + 1.73·83-s − 97-s + 1.73i·101-s + 2i·103-s + ⋯
L(s)  = 1  + 1.73i·5-s + i·7-s − 1.73·11-s − 1.99·25-s − i·31-s − 1.73·35-s + 1.73i·53-s − 2.99i·55-s + 73-s − 1.73i·77-s + 2i·79-s + 1.73·83-s − 97-s + 1.73i·101-s + 2i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8308065148\)
\(L(\frac12)\) \(\approx\) \(0.8308065148\)
\(L(1)\) 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 \( 1 - 1.73iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.73iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.962153402094828533162737104963, −9.090318317760965704052400355478, −7.971978144694336190983086303374, −7.55817730774435357563120269015, −6.56384369398420986721041059111, −5.85764038646417197966560838426, −5.10648996493238634169990687584, −3.73471513026515533627593127315, −2.65024315508821018248780196458, −2.38766893729123080263566463660, 0.60370191646727068006050159353, 1.90124744269204405741049955552, 3.34216952906879499504036469782, 4.46256558369872648350477891297, 5.01468287539011956956548146856, 5.71458768147761223305377234574, 6.98705215601696791934838476709, 7.85662416938244530563967150993, 8.324123945353074384211738989337, 9.147479059450892734500114128981

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