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Properties

Label 2-330-11.4-c1-0-4
Degree $2$
Conductor $330$
Sign $0.871 - 0.491i$
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 + (1.57 + 4.85i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (1.45 − 2.98i)11-s + 12-s + (−1.07 − 0.783i)13-s + (−1.57 + 4.85i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.627 + 0.455i)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 + (0.596 + 1.83i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.316·10-s + (0.437 − 0.899i)11-s + 0.288·12-s + (−0.299 − 0.217i)13-s + (−0.421 + 1.29i)14-s + (−0.0797 − 0.245i)15-s + (−0.202 + 0.146i)16-s + (−0.152 + 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.871 - 0.491i$
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: \((2,\ 330,\ (\ :1/2),\ 0.871 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98915 + 0.522067i\)
\(L(\frac12)\) \(\approx\) \(1.98915 + 0.522067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 + 2.98i)T \)
good7 \( 1 + (-1.57 - 4.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.07 + 0.783i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.627 - 0.455i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0242 - 0.0746i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.64T + 23T^{2} \)
29 \( 1 + (0.623 + 1.92i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.86 + 4.99i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.21 + 3.74i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.67 - 11.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.01T + 43T^{2} \)
47 \( 1 + (-2.03 + 6.24i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.36 + 6.08i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.64 + 5.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.03 + 2.20i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.588T + 67T^{2} \)
71 \( 1 + (-1.87 + 1.36i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.38 - 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.97 + 7.24i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.47 + 4.70i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + (-10.8 - 7.90i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.73924140638060705650626444910, −11.23073969144016948236051723156, −9.407813901985867790915664766526, −8.694061338990981362197754703197, −7.969670056346636271193572118618, −6.59238690341365714117578300894, −5.71863842663243279144528616977, −5.00792528956748352951612968798, −3.18634734243888465250882095544, −1.98779892826441129886439413558, 1.60176541926554238314642007021, 3.34868081297855086417754692471, 4.37464956348748461813659556921, 5.11214856758904833066181968326, 6.83023349371420132205938808010, 7.38085269956755726225985448304, 8.971913122651259948950964374881, 9.963071758475472807038460133819, 10.65472992346395770099234739827, 11.24431485563485061706457215240

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