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Properties

Label 2-330-33.32-c1-0-12
Degree $2$
Conductor $330$
Sign $0.977 - 0.211i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (1.25 + 1.19i)3-s + 4-s − i·5-s + (1.25 + 1.19i)6-s − 3.22i·7-s + 8-s + (0.166 + 2.99i)9-s − i·10-s + (2.73 + 1.87i)11-s + (1.25 + 1.19i)12-s − 0.712i·13-s − 3.22i·14-s + (1.19 − 1.25i)15-s + 16-s − 6.12·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.726 + 0.687i)3-s + 0.5·4-s − 0.447i·5-s + (0.513 + 0.485i)6-s − 1.22i·7-s + 0.353·8-s + (0.0553 + 0.998i)9-s − 0.316i·10-s + (0.825 + 0.564i)11-s + (0.363 + 0.343i)12-s − 0.197i·13-s − 0.863i·14-s + (0.307 − 0.324i)15-s + 0.250·16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41706 + 0.258433i\)
\(L(\frac12)\) \(\approx\) \(2.41706 + 0.258433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.25 - 1.19i)T \)
5 \( 1 + iT \)
11 \( 1 + (-2.73 - 1.87i)T \)
good7 \( 1 + 3.22iT - 7T^{2} \)
13 \( 1 + 0.712iT - 13T^{2} \)
17 \( 1 + 6.12T + 17T^{2} \)
19 \( 1 - 2.33iT - 19T^{2} \)
23 \( 1 - 3.03iT - 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + 9.55T + 31T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 - 0.761T + 41T^{2} \)
43 \( 1 + 5.79iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + 1.09iT - 53T^{2} \)
59 \( 1 - 1.95iT - 59T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 - 2.80T + 67T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−11.49995498396442145307898863566, −10.75006607863006792756121473172, −9.762293053421309167131688326410, −8.936998538071389698238354731172, −7.70240251558031936655004569838, −6.89442111739814472938160780410, −5.37838615318882172345813414196, −4.18972632182008154189291530485, −3.77403289996368841709465576137, −1.95204764389964242035112515087, 2.04150866178860465763316772635, 2.99408549748649944348884928185, 4.28220426282908288319696724914, 5.92008900532471792597129185760, 6.54327570846462333895177522508, 7.60588964382294450755990657529, 8.859774179328043417455377016349, 9.304858946115794850063193729846, 11.10700743579029359337814310527, 11.58767680479828257525116371269

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