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Properties

Label 2-12e3-9.4-c1-0-3
Degree $2$
Conductor $1728$
Sign $-0.0281 - 0.999i$
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  + (−1.18 + 2.05i)5-s + (−1.10 − 1.91i)7-s + (2.96 + 5.14i)11-s + (2.18 − 3.78i)13-s − 3.37·17-s + 3.72·19-s + (1.10 − 1.91i)23-s + (−0.313 − 0.543i)25-s + (−0.186 − 0.322i)29-s + (−4.83 + 8.36i)31-s + 5.24·35-s − 4·37-s + (0.5 − 0.866i)41-s + (2.96 + 5.14i)43-s + (1.10 + 1.91i)47-s + ⋯
L(s)  = 1  + (−0.530 + 0.918i)5-s + (−0.417 − 0.723i)7-s + (0.894 + 1.54i)11-s + (0.606 − 1.05i)13-s − 0.817·17-s + 0.854·19-s + (0.230 − 0.399i)23-s + (−0.0627 − 0.108i)25-s + (−0.0345 − 0.0598i)29-s + (−0.867 + 1.50i)31-s + 0.886·35-s − 0.657·37-s + (0.0780 − 0.135i)41-s + (0.452 + 0.783i)43-s + (0.161 + 0.279i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0281 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.0281 - 0.999i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.0281 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273968208\)
\(L(\frac12)\) \(\approx\) \(1.273968208\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.10 + 1.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.96 - 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.18 + 3.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 - 3.72T + 19T^{2} \)
23 \( 1 + (-1.10 + 1.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.186 + 0.322i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 - 8.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.96 - 5.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.10 - 1.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (5.17 - 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.55 - 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.17 - 8.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + (-4.83 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.04 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.25T + 89T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.580405856595787370542921073653, −8.797355606705064072820898675566, −7.69450717785158185865400611152, −7.00667302875125785568666711713, −6.72809916668814731900737512389, −5.47111582581822812145770485567, −4.35045674566762976168947889753, −3.63733701063192806089646621835, −2.76396737136100062443867475743, −1.27992393557980266644772216734, 0.53206111431234991727724713400, 1.85107608860105508847457713832, 3.33228499072756416399018815698, 3.98674837456652559194771318690, 5.02360077090318853860305588510, 5.97153266610858701863410499552, 6.52567430396992743242715644079, 7.67805819060572646869517091183, 8.585009003957240525281337532292, 9.077064647375805729642462420768

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