↑

Properties

Label 2-12e3-9.2-c2-0-3
Degree $2$
Conductor $1728$
Sign $-0.815 + 0.578i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.439 − 0.253i)5-s + (−6.44 + 11.1i)7-s + (−13.3 − 7.69i)11-s + (8.16 + 14.1i)13-s + 15.2i·17-s + 5.84·19-s + (21.4 − 12.3i)23-s + (−12.3 + 21.4i)25-s + (34.5 + 19.9i)29-s + (−18.1 − 31.4i)31-s + 6.54i·35-s − 6.15·37-s + (−33.5 + 19.3i)41-s + (−2.89 + 5.01i)43-s + (−45.8 − 26.4i)47-s + ⋯
L(s)  = 1  + (0.0879 − 0.0507i)5-s + (−0.921 + 1.59i)7-s + (−1.21 − 0.699i)11-s + (0.628 + 1.08i)13-s + 0.894i·17-s + 0.307·19-s + (0.931 − 0.537i)23-s + (−0.494 + 0.857i)25-s + (1.19 + 0.687i)29-s + (−0.586 − 1.01i)31-s + 0.187i·35-s − 0.166·37-s + (−0.817 + 0.471i)41-s + (−0.0673 + 0.116i)43-s + (−0.975 − 0.563i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2396974719\)
\(L(\frac12)\) \(\approx\) \(0.2396974719\)
\(L(2)\) 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 \( 1 + (-0.439 + 0.253i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (6.44 - 11.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (13.3 + 7.69i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.16 - 14.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 15.2iT - 289T^{2} \)
19 \( 1 - 5.84T + 361T^{2} \)
23 \( 1 + (-21.4 + 12.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-34.5 - 19.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (18.1 + 31.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 6.15T + 1.36e3T^{2} \)
41 \( 1 + (33.5 - 19.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.89 - 5.01i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (45.8 + 26.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 17.2iT - 2.80e3T^{2} \)
59 \( 1 + (-31.3 + 18.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-2.36 + 4.10i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.0 + 36.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.6iT - 5.04e3T^{2} \)
73 \( 1 + 134.T + 5.32e3T^{2} \)
79 \( 1 + (27.4 - 47.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (92.1 + 53.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 6.93iT - 7.92e3T^{2} \)
97 \( 1 + (-56.0 + 97.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.526179085578463066986932451764, −8.640031745297747304977092824077, −8.443487265398618163022374334681, −7.10843742695232373886761913995, −6.23037311236519019524226748630, −5.70371550003098176943436044912, −4.86644577910350112316808201535, −3.48738673319442599596691478899, −2.79903875312394232046703233480, −1.73440749322034631317528110159, 0.06860020956163257791018327977, 1.08672452034326946043047644811, 2.77004116409395862324245264163, 3.42068959641944536789733648589, 4.51861978263446392638723851348, 5.31883015548039089468640616529, 6.37514459407646502949610431243, 7.22371874733975061855986945201, 7.63589755482401585108518664095, 8.606962178367980802655669740884

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