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Properties

Label 2-12e3-9.7-c1-0-14
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: $\chi_{1728} (1153, \cdot )$
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

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

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