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Properties

Label 2-12e3-72.61-c1-0-12
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

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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: $\chi_{1728} (289, \cdot )$
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 \( 1 + (-9.84 + 17.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

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

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