Properties

Label 2-330-55.18-c1-0-8
Degree $2$
Conductor $330$
Sign $-0.132 + 0.991i$
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  + (−0.156 − 0.987i)2-s + (0.891 − 0.453i)3-s + (−0.951 + 0.309i)4-s + (1.66 − 1.48i)5-s + (−0.587 − 0.809i)6-s + (0.949 − 1.86i)7-s + (0.453 + 0.891i)8-s + (0.587 − 0.809i)9-s + (−1.73 − 1.41i)10-s + (−2.52 + 2.14i)11-s + (−0.707 + 0.707i)12-s + (2.76 − 0.437i)13-s + (−1.98 − 0.646i)14-s + (0.811 − 2.08i)15-s + (0.809 − 0.587i)16-s + (−1.45 − 0.230i)17-s + ⋯
L(s)  = 1  + (−0.110 − 0.698i)2-s + (0.514 − 0.262i)3-s + (−0.475 + 0.154i)4-s + (0.746 − 0.665i)5-s + (−0.239 − 0.330i)6-s + (0.358 − 0.704i)7-s + (0.160 + 0.315i)8-s + (0.195 − 0.269i)9-s + (−0.547 − 0.447i)10-s + (−0.761 + 0.647i)11-s + (−0.204 + 0.204i)12-s + (0.766 − 0.121i)13-s + (−0.531 − 0.172i)14-s + (0.209 − 0.538i)15-s + (0.202 − 0.146i)16-s + (−0.353 − 0.0559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ -0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03721 - 1.18464i\)
\(L(\frac12)\) \(\approx\) \(1.03721 - 1.18464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 + (-1.66 + 1.48i)T \)
11 \( 1 + (2.52 - 2.14i)T \)
good7 \( 1 + (-0.949 + 1.86i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (-2.76 + 0.437i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.45 + 0.230i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.188 - 0.579i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.38 + 1.38i)T + 23iT^{2} \)
29 \( 1 + (0.516 + 1.59i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.30 - 0.946i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.78 - 4.47i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (10.0 + 3.26i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (8.33 - 8.33i)T - 43iT^{2} \)
47 \( 1 + (-2.91 - 5.71i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.606 - 3.83i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-11.3 + 3.70i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.09 - 1.50i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.10 + 6.10i)T - 67iT^{2} \)
71 \( 1 + (9.05 - 6.58i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.99 - 5.09i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-2.25 - 1.63i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.12 - 13.4i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 1.07iT - 89T^{2} \)
97 \( 1 + (-8.60 + 1.36i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.27017448068401069462426160588, −10.26106244088264593898608502914, −9.651168477784024006998514974046, −8.525356022331491436418526452866, −7.87547634719166833081852549266, −6.50261556593851407553308232660, −5.09618218136168253867567431186, −4.09352533014869190763015258036, −2.53174956021685336983709977085, −1.28661500905936987212500279814, 2.18182635534860274313368798894, 3.53621276433266580068222007200, 5.15634813460494510791882596286, 5.95837462995855850707803737373, 7.01175790573329641672253180137, 8.238382327276894866284275904334, 8.826768390605521957678419436865, 9.875542342316208293210399931004, 10.69839035013439188990844997590, 11.67767026121455730514084091073

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