Properties

Label 2-330-11.4-c1-0-3
Degree $2$
Conductor $330$
Sign $0.0694 - 0.997i$
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 + (0.427 + 1.31i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (1.69 + 2.85i)11-s − 12-s + (1.30 + 0.951i)13-s + (−0.427 + 1.31i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−2 + 1.45i)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 + (0.161 + 0.496i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.316·10-s + (0.509 + 0.860i)11-s − 0.288·12-s + (0.363 + 0.263i)13-s + (−0.114 + 0.351i)14-s + (0.0797 + 0.245i)15-s + (−0.202 + 0.146i)16-s + (−0.485 + 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.0694 - 0.997i$
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: \((2,\ 330,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31152 + 1.22335i\)
\(L(\frac12)\) \(\approx\) \(1.31152 + 1.22335i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 2.85i)T \)
good7 \( 1 + (-0.427 - 1.31i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2 - 1.45i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.5 + 1.53i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.85T + 23T^{2} \)
29 \( 1 + (3 + 9.23i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.85 - 2.80i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.5 - 1.53i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.95 + 6.01i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (0.118 - 0.363i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.11 + 2.99i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.82 + 11.7i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.47 - 3.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (-11.0 + 8.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.14 + 6.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.61 + 2.62i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.4 - 7.60i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-2.38 - 1.73i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.90427621702599150495230950882, −11.03849552031325068031263252403, −9.821193641547063883339468762979, −9.059000763516615612636679150287, −8.036684186803731544453204665535, −6.71058518329990958362863256918, −5.85822915768378086444413794707, −4.79377737817970356363662792510, −3.91668272618697116520026353743, −2.21023128169050730130158582728, 1.26749903171290995414401825667, 2.85694669447096723450374379971, 4.10644673390485393505708395986, 5.51470695374629689945872025419, 6.34321403608521362482391903057, 7.32812808142253993421442338603, 8.545391467611668688541069298243, 9.665830364699202144792223007981, 10.84341622620299293182020799635, 11.25386980070950861578841292062

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