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Properties

Label 2-12e3-72.29-c0-0-0
Degree $2$
Conductor $1728$
Sign $0.422 - 0.906i$
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  + (−0.866 + 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s − 73-s + (−0.5 + 0.866i)97-s + 1.73·107-s + ⋯
L(s)  = 1  + (−0.866 + 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s − 73-s + (−0.5 + 0.866i)97-s + 1.73·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000322007\)
\(L(\frac12)\) \(\approx\) \(1.000322007\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.768568283645917151980432812103, −8.846224988193966430622710417297, −7.938566902550447214034421477236, −7.47399823976513068375033448673, −6.37993157883344568757058476937, −5.68819124423706306263499621154, −4.60331481726555117664409483617, −3.95101750871998102850184635250, −2.61756118153172588546874078105, −1.66403554509954787578173124639, 0.78312763341026005372339263250, 2.62225438757791549003331827602, 3.17077711068374732011850422639, 4.51249009048629713882919057816, 5.33414598472279731136179283812, 6.03248201438562069025797984885, 7.15722311472711143265453724016, 7.67084296676855900867471817619, 8.781278584208087241775730348652, 9.154737724619339469272197034332

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