Properties

Label 2-12e3-24.5-c2-0-0
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  − 9.03·5-s − 7.92·7-s + 1.07·11-s + 15.7i·13-s + 23.3i·17-s + 31.2i·19-s + 4.34i·23-s + 56.6·25-s + 0.634·29-s − 18.1·31-s + 71.6·35-s − 8.79i·37-s − 39.2i·41-s + 62.8i·43-s − 71.0i·47-s + ⋯
L(s)  = 1  − 1.80·5-s − 1.13·7-s + 0.0972·11-s + 1.20i·13-s + 1.37i·17-s + 1.64i·19-s + 0.188i·23-s + 2.26·25-s + 0.0218·29-s − 0.586·31-s + 2.04·35-s − 0.237i·37-s − 0.957i·41-s + 1.46i·43-s − 1.51i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.04726458922\)
\(L(\frac12)\) \(\approx\) \(0.04726458922\)
\(L(2)\) 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 \( 1 + 9.03T + 25T^{2} \)
7 \( 1 + 7.92T + 49T^{2} \)
11 \( 1 - 1.07T + 121T^{2} \)
13 \( 1 - 15.7iT - 169T^{2} \)
17 \( 1 - 23.3iT - 289T^{2} \)
19 \( 1 - 31.2iT - 361T^{2} \)
23 \( 1 - 4.34iT - 529T^{2} \)
29 \( 1 - 0.634T + 841T^{2} \)
31 \( 1 + 18.1T + 961T^{2} \)
37 \( 1 + 8.79iT - 1.36e3T^{2} \)
41 \( 1 + 39.2iT - 1.68e3T^{2} \)
43 \( 1 - 62.8iT - 1.84e3T^{2} \)
47 \( 1 + 71.0iT - 2.20e3T^{2} \)
53 \( 1 + 90.1T + 2.80e3T^{2} \)
59 \( 1 + 28.7T + 3.48e3T^{2} \)
61 \( 1 - 108. iT - 3.72e3T^{2} \)
67 \( 1 + 70.8iT - 4.48e3T^{2} \)
71 \( 1 - 14.2iT - 5.04e3T^{2} \)
73 \( 1 + 73.6T + 5.32e3T^{2} \)
79 \( 1 - 1.35T + 6.24e3T^{2} \)
83 \( 1 + 119.T + 6.88e3T^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + 83.0T + 9.40e3T^{2} \)
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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

−9.656226484655173728189735170702, −8.767891312514202214717528544357, −8.114224509601654743470228345547, −7.35283967934707309757753697260, −6.60863429071345284827603183610, −5.82277380198013486609279336344, −4.39080320081231086030606932034, −3.83578945144351995409585942545, −3.23370178852807362092643095258, −1.61248752741126687493382048614, 0.02178518514695568688705650196, 0.63904650626269700289834124216, 2.97637719688520554144346018519, 3.17745005970824193207271185441, 4.38485194968526927588886389960, 5.08583097130580450791140442811, 6.35186735738239657881208025138, 7.18228268307577434709582076566, 7.62340834314802156210062582876, 8.560783685392222245890702275866

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