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Properties

Label 2-12e3-8.3-c2-0-18
Degree $2$
Conductor $1728$
Sign $-0.965 - 0.258i$
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  + 8.36i·5-s + 2.43i·7-s + 2.41·11-s + 20.8i·13-s − 13.1·17-s + 32.7·19-s + 33.8i·23-s − 45.0·25-s − 34.7i·29-s + 33.7i·31-s − 20.4·35-s + 30.8i·37-s + 35.1·41-s + 27.9·43-s − 21.9i·47-s + ⋯
L(s)  = 1  + 1.67i·5-s + 0.348i·7-s + 0.219·11-s + 1.60i·13-s − 0.775·17-s + 1.72·19-s + 1.47i·23-s − 1.80·25-s − 1.19i·29-s + 1.08i·31-s − 0.583·35-s + 0.833i·37-s + 0.857·41-s + 0.649·43-s − 0.467i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.965 - 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.688395675\)
\(L(\frac12)\) \(\approx\) \(1.688395675\)
\(L(2)\) 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 \( 1 - 8.36iT - 25T^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 - 2.41T + 121T^{2} \)
13 \( 1 - 20.8iT - 169T^{2} \)
17 \( 1 + 13.1T + 289T^{2} \)
19 \( 1 - 32.7T + 361T^{2} \)
23 \( 1 - 33.8iT - 529T^{2} \)
29 \( 1 + 34.7iT - 841T^{2} \)
31 \( 1 - 33.7iT - 961T^{2} \)
37 \( 1 - 30.8iT - 1.36e3T^{2} \)
41 \( 1 - 35.1T + 1.68e3T^{2} \)
43 \( 1 - 27.9T + 1.84e3T^{2} \)
47 \( 1 + 21.9iT - 2.20e3T^{2} \)
53 \( 1 + 26.3iT - 2.80e3T^{2} \)
59 \( 1 + 93.3T + 3.48e3T^{2} \)
61 \( 1 - 27.7iT - 3.72e3T^{2} \)
67 \( 1 + 9.97T + 4.48e3T^{2} \)
71 \( 1 - 64.5iT - 5.04e3T^{2} \)
73 \( 1 + 45.5T + 5.32e3T^{2} \)
79 \( 1 + 82.8iT - 6.24e3T^{2} \)
83 \( 1 - 59.4T + 6.88e3T^{2} \)
89 \( 1 + 139.T + 7.92e3T^{2} \)
97 \( 1 - 150.T + 9.40e3T^{2} \)
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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

−9.539994999795262136804278728574, −8.891119965047269554454090028634, −7.59851331026563568944896110596, −7.13621947363342135118582121346, −6.41053568813474680195795372596, −5.65178447430282542018653760890, −4.41038282971980563567649917642, −3.46724982479596344731165872297, −2.65484497093404650396267440707, −1.62713142439393031249718112998, 0.49225840686527065640907329717, 1.16733195920597417685897709848, 2.65085370093428803966547081304, 3.85798956069382450776797280648, 4.73467800747084105996788941267, 5.37588388863407470082602723534, 6.16422843976343250798120220195, 7.50111094034963411670146685459, 7.933625526839202070258787809577, 8.954165949176458491217424272003

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