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Properties

Label 2-12e3-24.5-c2-0-53
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  + 0.519·5-s + 10.4·7-s − 14.1·11-s + 5.39i·13-s − 24.9i·17-s + 13.3i·19-s − 41.1i·23-s − 24.7·25-s + 7.85·29-s − 26.1·31-s + 5.40·35-s + 1.53i·37-s − 21.3i·41-s − 64.1i·43-s + 19.8i·47-s + ⋯
L(s)  = 1  + 0.103·5-s + 1.48·7-s − 1.28·11-s + 0.414i·13-s − 1.46i·17-s + 0.701i·19-s − 1.78i·23-s − 0.989·25-s + 0.270·29-s − 0.844·31-s + 0.154·35-s + 0.0415i·37-s − 0.520i·41-s − 1.49i·43-s + 0.422i·47-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

\(L(\frac{3}{2})\) \(\approx\) \(1.460506606\)
\(L(\frac12)\) \(\approx\) \(1.460506606\)
\(L(2)\) 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 \( 1 - 0.519T + 25T^{2} \)
7 \( 1 - 10.4T + 49T^{2} \)
11 \( 1 + 14.1T + 121T^{2} \)
13 \( 1 - 5.39iT - 169T^{2} \)
17 \( 1 + 24.9iT - 289T^{2} \)
19 \( 1 - 13.3iT - 361T^{2} \)
23 \( 1 + 41.1iT - 529T^{2} \)
29 \( 1 - 7.85T + 841T^{2} \)
31 \( 1 + 26.1T + 961T^{2} \)
37 \( 1 - 1.53iT - 1.36e3T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 + 64.1iT - 1.84e3T^{2} \)
47 \( 1 - 19.8iT - 2.20e3T^{2} \)
53 \( 1 + 68.6T + 2.80e3T^{2} \)
59 \( 1 - 67.8T + 3.48e3T^{2} \)
61 \( 1 + 58.6iT - 3.72e3T^{2} \)
67 \( 1 - 56.1iT - 4.48e3T^{2} \)
71 \( 1 + 107. iT - 5.04e3T^{2} \)
73 \( 1 - 7.73T + 5.32e3T^{2} \)
79 \( 1 - 64.3T + 6.24e3T^{2} \)
83 \( 1 - 125.T + 6.88e3T^{2} \)
89 \( 1 - 59.6iT - 7.92e3T^{2} \)
97 \( 1 + 138.T + 9.40e3T^{2} \)
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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

−8.790044669439171763801553618634, −8.018632401296901823239945689406, −7.55256181557910236732720253013, −6.55171142845388763401201617527, −5.35090017020595433694521822449, −4.97261273259440702597472234120, −4.00031711967311038023609678481, −2.60657051522855383190446121104, −1.86084229739213613359377810496, −0.37071587942355454235664701862, 1.35636302717180701318905344389, 2.22798262820290052660190342489, 3.45878467101760420825636247855, 4.53331958880345318234268708071, 5.33215458786281793451718036827, 5.89441159931784388969617039625, 7.21025284074308085522348160079, 8.043748946304962456879250132191, 8.192413045243281098029266572270, 9.396905787225505400882839768730

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