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Properties

Label 2-330-11.9-c1-0-2
Degree $2$
Conductor $330$
Sign $0.605 - 0.795i$
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  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)6-s + (1 + 0.726i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s − 0.999·10-s + (2.54 + 2.12i)11-s − 0.999·12-s + (0.5 − 1.53i)13-s + (−1 + 0.726i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (1.19 + 3.66i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (0.126 + 0.388i)6-s + (0.377 + 0.274i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s − 0.316·10-s + (0.767 + 0.641i)11-s − 0.288·12-s + (0.138 − 0.426i)13-s + (−0.267 + 0.194i)14-s + (0.208 + 0.151i)15-s + (0.0772 + 0.237i)16-s + (0.288 + 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29796 + 0.643471i\)
\(L(\frac12)\) \(\approx\) \(1.29796 + 0.643471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-2.54 - 2.12i)T \)
good7 \( 1 + (-1 - 0.726i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.5 + 1.53i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.19 - 3.66i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.618 + 0.449i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + (-4.92 - 3.57i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.881 + 2.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.73 + 3.44i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.38 + 1.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.85T + 43T^{2} \)
47 \( 1 + (11.0 - 8.00i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.61 + 8.05i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.118 + 0.0857i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.472 + 1.45i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.32T + 67T^{2} \)
71 \( 1 + (4.76 + 14.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.61 - 2.62i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.95 + 9.09i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.472 + 1.45i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 5.70T + 89T^{2} \)
97 \( 1 + (-1.32 + 4.08i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.80082111808190422003410493220, −10.60636310665857160236405552443, −9.697732682725052342584789856252, −8.730938215088400762705071113119, −7.932017826917080087028689075262, −6.94708574127554820496289051696, −6.12557423969898956619024759948, −4.83009411235103003500378200871, −3.41017619811020294111524919565, −1.70521262834388830458715638015, 1.33458308038644648857458172379, 2.98131331046038323149206133000, 4.15327074577793032944197096572, 5.16031535692533066696052301575, 6.70277207717061233898813954422, 8.023979738669964766638285676136, 8.797198146770609627752572496224, 9.575369532422150682130552621457, 10.45145236598896759024087876349, 11.51526562243212151231664093717

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