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Properties

Label 2-12e3-9.4-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: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.0281 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8523314642\)
\(L(\frac12)\) \(\approx\) \(0.8523314642\)
\(L(\frac{3}{2})\) 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

−8.832088882513997573137717708117, −8.309150881746163748665824610752, −7.75689414133159750316029162605, −6.64334797744975377191447887031, −5.88526322525902696236269554356, −5.22051277163033795205196352603, −3.88438407513076570682291826669, −3.09130314457877190594183750976, −2.25215757902240464830299626602, −0.32848739650209368152553800138, 1.32239604121223499014212463136, 2.37341402824408821246459674337, 4.01367965359225674701737403973, 4.52877643335774378167821391964, 5.07961504140380808389855638023, 6.58654279509145031750248512556, 7.06230045688488078189120937455, 8.159772008952282944049098241881, 8.501883796077650282755207106888, 9.483858903815552291283351715346

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