↑

Properties

Label 2-12e3-72.13-c1-0-11
Degree $2$
Conductor $1728$
Sign $0.983 + 0.178i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.71 − 2.72i)11-s + 1.89·17-s + 8.34i·19-s + (−2.5 − 4.33i)25-s + (6.39 − 11.0i)41-s + (−2.03 + 1.17i)43-s + (3.5 − 6.06i)49-s + (8.00 + 4.62i)59-s + (12.4 + 7.17i)67-s + 13.6·73-s + (15.5 − 9i)83-s − 18·89-s + (9.84 + 17.0i)97-s + 20.1i·107-s + (9 − 15.5i)113-s + ⋯
L(s)  = 1  + (1.42 − 0.821i)11-s + 0.460·17-s + 1.91i·19-s + (−0.5 − 0.866i)25-s + (0.999 − 1.73i)41-s + (−0.310 + 0.179i)43-s + (0.5 − 0.866i)49-s + (1.04 + 0.601i)59-s + (1.51 + 0.876i)67-s + 1.60·73-s + (1.71 − 0.987i)83-s − 1.90·89-s + (0.999 + 1.73i)97-s + 1.94i·107-s + (0.846 − 1.46i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.983 + 0.178i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892499502\)
\(L(\frac12)\) \(\approx\) \(1.892499502\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.71 + 2.72i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 - 8.34iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-6.39 + 11.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.03 - 1.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-8.00 - 4.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.4 - 7.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.5 + 9i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-9.84 - 17.0i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.278426433840160640163180356644, −8.484529977280366405662747691919, −7.85097789841025182651675346139, −6.80966303341533203804417253995, −6.05805343434759514351418948224, −5.41141847455468167099477804648, −3.96675960348124341205227578511, −3.64931932962850526573371533933, −2.16199188911955429496958102201, −0.957337682443293058641963987031, 1.04733544284154866620138805439, 2.27592102628779514045667445901, 3.45517300963357768435460630472, 4.38801406965901878104815434488, 5.15127655986276964099660240115, 6.29686365088456265310089449610, 6.92529325762479723000949985172, 7.65737970404541939852462791883, 8.687123945721177898316764239609, 9.475376401828339331605410661949

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