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Properties

Label 2-12e3-8.5-c3-0-27
Degree $2$
Conductor $1728$
Sign $0.258 - 0.965i$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·7-s + 91.7i·13-s − 163i·19-s + 125·25-s + 155.·31-s − 313. i·37-s + 520i·43-s − 340·49-s + 625. i·61-s + 127i·67-s − 919·73-s − 219.·79-s − 159i·91-s + 523·97-s + 1.06e3·103-s + ⋯
L(s)  = 1  − 0.0935·7-s + 1.95i·13-s − 1.96i·19-s + 25-s + 0.903·31-s − 1.39i·37-s + 1.84i·43-s − 0.991·49-s + 1.31i·61-s + 0.231i·67-s − 1.47·73-s − 0.313·79-s − 0.183i·91-s + 0.547·97-s + 1.01·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.728805476\)
\(L(\frac12)\) \(\approx\) \(1.728805476\)
\(L(\frac{5}{2})\) 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 \( 1 - 125T^{2} \)
7 \( 1 + 1.73T + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 - 91.7iT - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 163iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 155.T + 2.97e4T^{2} \)
37 \( 1 + 313. iT - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 520iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 - 625. iT - 2.26e5T^{2} \)
67 \( 1 - 127iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 919T + 3.89e5T^{2} \)
79 \( 1 + 219.T + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 523T + 9.12e5T^{2} \)
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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

−9.115678480342251994168988752695, −8.551914245403315444275934299582, −7.32513703611446640468079055422, −6.81046765058969886622343914944, −6.05974752039640658702876936484, −4.75241159171612797542042683976, −4.39091997758369105162979624373, −3.07110242143556463212962473125, −2.16239078533963183469376191596, −0.947306926521746880504357780099, 0.43778951983019546147670093414, 1.56151425800261587269196068902, 2.92264084675023507110674358166, 3.55231451553519128178799876089, 4.76658492051157705290141707712, 5.59499995524108001055050333564, 6.26575726033708255830636963305, 7.30865488372189199345028612002, 8.151302240667971361978221301448, 8.518478080595025320302420430536

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