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Properties

Label 2-12e3-8.5-c3-0-74
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  − 12.4i·5-s + 12.1·7-s − 21.6i·11-s − 15.5i·13-s + 64.8·17-s − 49i·19-s + 62.4·23-s − 31·25-s + 24.9i·29-s + 24.2·31-s − 151. i·35-s − 102. i·37-s + 346.·41-s − 260i·43-s + 362.·47-s + ⋯
L(s)  = 1  − 1.11i·5-s + 0.654·7-s − 0.592i·11-s − 0.332i·13-s + 0.925·17-s − 0.591i·19-s + 0.566·23-s − 0.247·25-s + 0.159i·29-s + 0.140·31-s − 0.731i·35-s − 0.454i·37-s + 1.31·41-s − 0.922i·43-s + 1.12·47-s + ⋯

Functional equation

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

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

\(L(2)\) \(\approx\) \(2.426720889\)
\(L(\frac12)\) \(\approx\) \(2.426720889\)
\(L(\frac{5}{2})\) 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 \( 1 + 12.4iT - 125T^{2} \)
7 \( 1 - 12.1T + 343T^{2} \)
11 \( 1 + 21.6iT - 1.33e3T^{2} \)
13 \( 1 + 15.5iT - 2.19e3T^{2} \)
17 \( 1 - 64.8T + 4.91e3T^{2} \)
19 \( 1 + 49iT - 6.85e3T^{2} \)
23 \( 1 - 62.4T + 1.21e4T^{2} \)
29 \( 1 - 24.9iT - 2.43e4T^{2} \)
31 \( 1 - 24.2T + 2.97e4T^{2} \)
37 \( 1 + 102. iT - 5.06e4T^{2} \)
41 \( 1 - 346.T + 6.89e4T^{2} \)
43 \( 1 + 260iT - 7.95e4T^{2} \)
47 \( 1 - 362.T + 1.03e5T^{2} \)
53 \( 1 - 574. iT - 1.48e5T^{2} \)
59 \( 1 + 324. iT - 2.05e5T^{2} \)
61 \( 1 + 174. iT - 2.26e5T^{2} \)
67 \( 1 - 241iT - 3.00e5T^{2} \)
71 \( 1 + 249.T + 3.57e5T^{2} \)
73 \( 1 - 353T + 3.89e5T^{2} \)
79 \( 1 + 5.19T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 - 800.T + 7.04e5T^{2} \)
97 \( 1 - 1.11e3T + 9.12e5T^{2} \)
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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

−8.851373682243011197428597721335, −7.968984002771436286787960110809, −7.38130261362764698157232846354, −6.13391221024763669334424526808, −5.33225446970069485714742068646, −4.76002236793639750694410423367, −3.75704696987652314269364807567, −2.62246491031474461614997705540, −1.30164215758602349224721084545, −0.59408886306051266162699682845, 1.16632989188790747734917891639, 2.27347113409811667818315276527, 3.18487018502990446661218229121, 4.17010094957200634596774006849, 5.11220373964468929302143564686, 6.05953166886675636852222684633, 6.87963034100764351797908598227, 7.57404958858125680201451141470, 8.219936926477994468565828223754, 9.313763023540190046937625557542

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