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Properties

Label 2-12e3-36.31-c2-0-21
Degree $2$
Conductor $1728$
Sign $0.640 - 0.768i$
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  + (−3.31 + 5.73i)5-s + (−8.46 + 4.88i)7-s + (10.0 − 5.80i)11-s + (8.93 − 15.4i)13-s − 2.37·17-s − 14.0i·19-s + (35.9 + 20.7i)23-s + (−9.43 − 16.3i)25-s + (−5.68 − 9.85i)29-s + (−18.0 − 10.4i)31-s − 64.7i·35-s − 35.7·37-s + (2.62 − 4.55i)41-s + (54.4 − 31.4i)43-s + (−4.28 + 2.47i)47-s + ⋯
L(s)  = 1  + (−0.662 + 1.14i)5-s + (−1.20 + 0.698i)7-s + (0.914 − 0.528i)11-s + (0.687 − 1.19i)13-s − 0.139·17-s − 0.738i·19-s + (1.56 + 0.902i)23-s + (−0.377 − 0.653i)25-s + (−0.196 − 0.339i)29-s + (−0.581 − 0.335i)31-s − 1.85i·35-s − 0.965·37-s + (0.0641 − 0.111i)41-s + (1.26 − 0.731i)43-s + (−0.0911 + 0.0526i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.640 - 0.768i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.640 - 0.768i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.479163049\)
\(L(\frac12)\) \(\approx\) \(1.479163049\)
\(L(2)\) 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 \( 1 + (3.31 - 5.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (8.46 - 4.88i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.0 + 5.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.93 + 15.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 2.37T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-35.9 - 20.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (5.68 + 9.85i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (18.0 + 10.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 35.7T + 1.36e3T^{2} \)
41 \( 1 + (-2.62 + 4.55i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-54.4 + 31.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (4.28 - 2.47i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 75.7T + 2.80e3T^{2} \)
59 \( 1 + (-50.3 - 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.93 + 10.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 - 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 46.4iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + (18.0 - 10.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (24.4 - 14.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 73.0T + 7.92e3T^{2} \)
97 \( 1 + (-71.1 - 123. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.088472763273953251584500355285, −8.679471711304319509092264558569, −7.39723783760333373731275755478, −6.94292178054469432978241722252, −6.07678282728684526561891756533, −5.43345546999974190255841073497, −3.82346293919948490983869754758, −3.31305889775713543430505037574, −2.60173087509471537641655157478, −0.75049583103660921942832770521, 0.62223691732670770157375763182, 1.61209278386186234806112718698, 3.29942540513124194009142602075, 4.10006132173133668442110105051, 4.60244781395857930331283790870, 5.86264812217211015203714421477, 6.85379904273355350041331290051, 7.17846513430844773537585396953, 8.528574951676218401905453470373, 8.959120371614516380570183427653

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