Properties

Label 2-12e3-9.5-c2-0-6
Degree $2$
Conductor $1728$
Sign $0.766 - 0.642i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.39 − 1.96i)5-s + (−6.39 − 11.0i)7-s + (−14.2 + 8.25i)11-s + (−1.39 + 2.42i)13-s + 2.54i·17-s − 21.5·19-s + (−2.60 − 1.50i)23-s + (−4.79 − 8.31i)25-s + (−13.1 + 7.61i)29-s + (13.6 − 23.5i)31-s + 50.2i·35-s + 10.4·37-s + (34.5 + 19.9i)41-s + (17.0 + 29.6i)43-s + (−58.1 + 33.6i)47-s + ⋯
L(s)  = 1  + (−0.679 − 0.392i)5-s + (−0.914 − 1.58i)7-s + (−1.29 + 0.750i)11-s + (−0.107 + 0.186i)13-s + 0.149i·17-s − 1.13·19-s + (−0.113 − 0.0652i)23-s + (−0.191 − 0.332i)25-s + (−0.455 + 0.262i)29-s + (0.438 − 0.759i)31-s + 1.43i·35-s + 0.281·37-s + (0.841 + 0.485i)41-s + (0.397 + 0.688i)43-s + (−1.23 + 0.714i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4817063744\)
\(L(\frac12)\) \(\approx\) \(0.4817063744\)
\(L(2)\) 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 \( 1 + (3.39 + 1.96i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (6.39 + 11.0i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.2 - 8.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.39 - 2.42i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 2.54iT - 289T^{2} \)
19 \( 1 + 21.5T + 361T^{2} \)
23 \( 1 + (2.60 + 1.50i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13.1 - 7.61i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-13.6 + 23.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 10.4T + 1.36e3T^{2} \)
41 \( 1 + (-34.5 - 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-17.0 - 29.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (58.1 - 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-5.29 - 3.05i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-20.3 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.4 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 52.8iT - 5.04e3T^{2} \)
73 \( 1 - 68.7T + 5.32e3T^{2} \)
79 \( 1 + (-12.7 - 22.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-52.0 + 30.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.62iT - 7.92e3T^{2} \)
97 \( 1 + (-50.7 - 87.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.428794280761766845727361731072, −8.050251650864724049198411308963, −7.83327936376628648674196482148, −6.88649629236398920857790156436, −6.20390789602903905188883976657, −4.82388618778976018918412192627, −4.27204452873666910539818769149, −3.44749989445467143541125607870, −2.22047351622039785182921293633, −0.62497047715582913213327895865, 0.20772264429018792308871420732, 2.31103504013228149963099954614, 2.92046249838482300599648958405, 3.81717122987854392817187423575, 5.16171857271938035107179669192, 5.79689869101160703263901050923, 6.54965055892769027593061059423, 7.56042990914203305332323579852, 8.343371887337811970464914160458, 8.910880372004163638036437171633

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