Properties

Label 2-12e3-24.5-c2-0-51
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  + 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

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

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: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.965 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.545730556\)
\(L(\frac12)\) \(\approx\) \(3.545730556\)
\(L(2)\) 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

−8.955366101049731893494503849884, −8.765450803543300920676574445751, −7.21082571130081161291715563851, −6.83739774337610649317374101420, −5.73394138172106231903458615501, −5.06702098834247790416362784091, −4.40567611681201968228028150226, −2.69689336369473995072093153351, −2.06031555436571486944038800524, −1.04409106303624737561412387602, 1.29664738438863520573773053562, 1.87363262419832595829328620571, 3.00254054914550586804466394630, 4.30140158748253903181324226705, 5.34304672372498579588583676282, 5.84001440184114325333301349285, 6.49117555716042181687816892039, 7.87830955966058715996398007191, 8.260474458012045086294736769752, 9.231638819913670206516554428213

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