Properties

Label 2-12e3-3.2-c2-0-27
Degree $2$
Conductor $1728$
Sign $-i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.37i·5-s + 1.26·7-s + 5.82i·11-s + 10.4·13-s − 18.3i·17-s + 20.8·19-s + 20.4i·23-s − 29.4·25-s + 11.1i·29-s + 61.3·31-s + 9.34i·35-s + 38.4·37-s + 33.0i·41-s + 49.3·43-s − 21.5i·47-s + ⋯
L(s)  = 1  + 1.47i·5-s + 0.180·7-s + 0.529i·11-s + 0.806·13-s − 1.07i·17-s + 1.09·19-s + 0.890i·23-s − 1.17·25-s + 0.385i·29-s + 1.97·31-s + 0.266i·35-s + 1.03·37-s + 0.807i·41-s + 1.14·43-s − 0.459i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.213056883\)
\(L(\frac12)\) \(\approx\) \(2.213056883\)
\(L(2)\) 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 \( 1 - 7.37iT - 25T^{2} \)
7 \( 1 - 1.26T + 49T^{2} \)
11 \( 1 - 5.82iT - 121T^{2} \)
13 \( 1 - 10.4T + 169T^{2} \)
17 \( 1 + 18.3iT - 289T^{2} \)
19 \( 1 - 20.8T + 361T^{2} \)
23 \( 1 - 20.4iT - 529T^{2} \)
29 \( 1 - 11.1iT - 841T^{2} \)
31 \( 1 - 61.3T + 961T^{2} \)
37 \( 1 - 38.4T + 1.36e3T^{2} \)
41 \( 1 - 33.0iT - 1.68e3T^{2} \)
43 \( 1 - 49.3T + 1.84e3T^{2} \)
47 \( 1 + 21.5iT - 2.20e3T^{2} \)
53 \( 1 + 77.5iT - 2.80e3T^{2} \)
59 \( 1 - 25.7iT - 3.48e3T^{2} \)
61 \( 1 - 55.8T + 3.72e3T^{2} \)
67 \( 1 + 91.0T + 4.48e3T^{2} \)
71 \( 1 - 114. iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + 8.21T + 6.24e3T^{2} \)
83 \( 1 - 150. iT - 6.88e3T^{2} \)
89 \( 1 + 118. iT - 7.92e3T^{2} \)
97 \( 1 + 72.8T + 9.40e3T^{2} \)
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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

−9.665203229837728585474680109153, −8.440543589357136433587284435100, −7.56715005935015436582179736828, −7.00769717686139862543116558394, −6.26109380956074037465795980232, −5.34352541322195692620100489523, −4.27219451036568217959340501878, −3.19184949983705986024491780085, −2.60117490676669736400016079068, −1.17231258139626588283554867587, 0.70077206306656760896448132070, 1.45433132206426546669395842613, 2.91145048071967505700216232786, 4.13990131781773612160498897163, 4.69279587243954653746301367944, 5.78053926015889214845700990799, 6.23383703270615835117634527603, 7.63167436125054059526820064736, 8.311640705540671836434393680410, 8.789877124293276362711034536097

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