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Properties

Label 2-12e3-36.23-c1-0-10
Degree $2$
Conductor $1728$
Sign $0.766 - 0.642i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  + (1.5 + 0.866i)5-s + (2.59 − 1.5i)7-s + (2.59 + 4.5i)11-s + (0.5 − 0.866i)13-s + 3.46i·17-s + 6i·19-s + (2.59 − 4.5i)23-s + (−1 − 1.73i)25-s + (−7.5 + 4.33i)29-s + (2.59 + 1.5i)31-s + 5.19·35-s + 4·37-s + (−4.5 − 2.59i)41-s + (−2.59 + 1.5i)43-s + (2.59 + 4.5i)47-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (0.981 − 0.566i)7-s + (0.783 + 1.35i)11-s + (0.138 − 0.240i)13-s + 0.840i·17-s + 1.37i·19-s + (0.541 − 0.938i)23-s + (−0.200 − 0.346i)25-s + (−1.39 + 0.804i)29-s + (0.466 + 0.269i)31-s + 0.878·35-s + 0.657·37-s + (−0.702 − 0.405i)41-s + (−0.396 + 0.228i)43-s + (0.378 + 0.656i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.289581495\)
\(L(\frac12)\) \(\approx\) \(2.289581495\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-2.59 + 4.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.5 - 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.59 - 1.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.59 - 1.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.59 - 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (12.9 - 7.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.612344284980295743843025763084, −8.514364276707030169659894748434, −7.85611334984333022206915637276, −6.96603237533360795245617375977, −6.30697054988178164174554684047, −5.30700417806270389676596619527, −4.41076168718422426365874978234, −3.62998697381350297543096464944, −2.10349967106434130794708881196, −1.45263796481945951255026305622, 0.961778340734451286013694037010, 2.05591758248898705445138687247, 3.17082285219244111478306867158, 4.35214265314807212888193634006, 5.33107360863464952465090785533, 5.77414362601839007072782100136, 6.80401169135708968153948522971, 7.72928440923653514022281016746, 8.675501905553982210426839155071, 9.133856397679323348172098771863

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