Properties

Label 2-330-11.4-c1-0-6
Degree $2$
Conductor $330$
Sign $0.893 + 0.449i$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−1.38 − 4.27i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (3.28 − 0.458i)11-s + 12-s + (1.88 + 1.37i)13-s + (1.38 − 4.27i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (4.17 − 3.03i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.330 − 0.239i)6-s + (−0.524 − 1.61i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.316·10-s + (0.990 − 0.138i)11-s + 0.288·12-s + (0.523 + 0.380i)13-s + (0.370 − 1.14i)14-s + (−0.0797 − 0.245i)15-s + (−0.202 + 0.146i)16-s + (1.01 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89150 - 0.448661i\)
\(L(\frac12)\) \(\approx\) \(1.89150 - 0.448661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.28 + 0.458i)T \)
good7 \( 1 + (1.38 + 4.27i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.88 - 1.37i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.17 + 3.03i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.85 - 5.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 + (-2.34 - 7.21i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.897 - 0.651i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.45 - 7.54i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.95 + 9.10i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.0785T + 43T^{2} \)
47 \( 1 + (-0.896 + 2.76i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.602 + 0.437i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.64 + 5.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (12.5 - 9.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.67T + 67T^{2} \)
71 \( 1 + (7.72 - 5.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.38 - 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.42 - 3.21i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.47 + 4.70i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 7.41T + 89T^{2} \)
97 \( 1 + (4.64 + 3.37i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.88196166091087776157442299325, −10.53721140887491313753168132655, −9.705427677295878922589662732696, −8.457285271518540783258620781839, −7.48920237599776389259229859091, −6.64039300591870941004690387247, −5.85638354578387654365025493444, −4.24184685275114025654145310378, −3.45069244898643689214877258137, −1.37708381585938722577075507860, 2.17388353935776309579349552043, 3.25764225835461338691830439947, 4.45796871340449753612299988711, 6.01137229792507312362471569686, 6.09813740554824802522174247384, 8.068929825276068611491805783717, 9.196883924179442956065843836169, 9.721170226067197222530317424978, 10.78587667161625976287482414387, 11.80934954106874309271062605033

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