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Properties

Label 2-12e3-9.7-c1-0-7
Degree $2$
Conductor $1728$
Sign $-0.118 - 0.993i$
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  + (1.68 + 2.92i)5-s + (−0.686 + 1.18i)7-s + (0.5 − 0.866i)11-s + (2.68 + 4.65i)13-s − 0.372·17-s + 6.37·19-s + (−2.68 − 4.65i)23-s + (−3.18 + 5.51i)25-s + (−0.686 + 1.18i)29-s + (0.313 + 0.543i)31-s − 4.62·35-s + 2.74·37-s + (−0.127 − 0.221i)41-s + (−4.87 + 8.43i)43-s + (0.686 − 1.18i)47-s + ⋯
L(s)  = 1  + (0.754 + 1.30i)5-s + (−0.259 + 0.449i)7-s + (0.150 − 0.261i)11-s + (0.745 + 1.29i)13-s − 0.0902·17-s + 1.46·19-s + (−0.560 − 0.970i)23-s + (−0.637 + 1.10i)25-s + (−0.127 + 0.220i)29-s + (0.0563 + 0.0976i)31-s − 0.782·35-s + 0.451·37-s + (−0.0199 − 0.0345i)41-s + (−0.743 + 1.28i)43-s + (0.100 − 0.173i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.118 - 0.993i$
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: \((2,\ 1728,\ (\ :1/2),\ -0.118 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920913175\)
\(L(\frac12)\) \(\approx\) \(1.920913175\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 2.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.686 - 1.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.68 - 4.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.372T + 17T^{2} \)
19 \( 1 - 6.37T + 19T^{2} \)
23 \( 1 + (2.68 + 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.686 - 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.313 - 0.543i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.74T + 37T^{2} \)
41 \( 1 + (0.127 + 0.221i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 - 8.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.686 + 1.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.87 + 6.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (0.313 - 0.543i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.68 + 13.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-4.87 + 8.43i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.522617807679799258885827002311, −8.960896456263568994143223337757, −7.888628561919728125889232846254, −6.92709828322135094663247849129, −6.33174557438577982036751472002, −5.79957238504614618419366232888, −4.55411492622459458528816860173, −3.40079407867030454196303086427, −2.66148492171291933269431175891, −1.57649393294833677303553009922, 0.76142391979438359876843229339, 1.69579933663929288273973558451, 3.17692316946572100742291090213, 4.09711405489823836825393024398, 5.31756264298255985854964911085, 5.54206363425079465280401965759, 6.63977242367379124203311791472, 7.72400231682457338403554153540, 8.289411428632067002791775783718, 9.282544760701378060299121350416

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