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Properties

Label 2-12e3-9.4-c1-0-19
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: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.118 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.835229019\)
\(L(\frac12)\) \(\approx\) \(1.835229019\)
\(L(\frac{3}{2})\) 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

−8.918091371746900661686670536663, −8.481586824634864073309496604583, −7.84907256981512769010277137304, −6.39295874828119662267234050978, −5.85026577377556401600928006101, −5.04457379243453450038923724663, −4.32413406676027606526195707345, −2.96003692259833951661411185933, −1.85990343991887354819769929137, −0.68867468359687802497943078473, 1.62782332311471753126519795548, 2.47242067265729120143241212922, 3.63575021234756299602021324774, 4.45340516290197096065073738055, 5.67185000483955682735877541030, 6.50817676675377657168449269564, 6.93584009653873755053562014288, 7.82359257588570266786813233791, 8.899425255830466906361655749960, 9.548358102663763709630701536492

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