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Properties

Label 2-12e3-24.5-c2-0-18
Degree $2$
Conductor $1728$
Sign $-0.258 - 0.965i$
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  + 8.66·7-s + 12.1i·13-s + 37i·19-s − 25·25-s − 41.5·31-s + 12.1i·37-s − 22i·43-s + 26.0·49-s + 112. i·61-s − 13i·67-s + 143·73-s − 157.·79-s + 105. i·91-s + 169·97-s − 133.·103-s + ⋯
L(s)  = 1  + 1.23·7-s + 0.932i·13-s + 1.94i·19-s − 25-s − 1.34·31-s + 0.327i·37-s − 0.511i·43-s + 0.530·49-s + 1.84i·61-s − 0.194i·67-s + 1.95·73-s − 1.99·79-s + 1.15i·91-s + 1.74·97-s − 1.29·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.659356049\)
\(L(\frac12)\) \(\approx\) \(1.659356049\)
\(L(2)\) 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 \( 1 + 25T^{2} \)
7 \( 1 - 8.66T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 12.1iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 37iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 41.5T + 961T^{2} \)
37 \( 1 - 12.1iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 112. iT - 3.72e3T^{2} \)
67 \( 1 + 13iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 143T + 5.32e3T^{2} \)
79 \( 1 + 157.T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 169T + 9.40e3T^{2} \)
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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

−9.330068478305917262989331501683, −8.470884402841921495215164101982, −7.85063434356406380423594201966, −7.13143682226981523988390583062, −6.02181968249177398257891592800, −5.34489558492863087586462195362, −4.32693539285892145474069977188, −3.65323958240326384190288519941, −2.10085236640920717460256251327, −1.43799875889598357498284471659, 0.42567931303194093153506992793, 1.73056997818829243686017114247, 2.75700825166510502139121451917, 3.92184456919840532760438887052, 4.93221464760347719078475282612, 5.42879658043597664997773762007, 6.55136480110777381661525615130, 7.50516612508426012461023307860, 8.021253210492560808635042855281, 8.885804028338135961990412782087

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