Properties

Degree $2$
Conductor $330$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 11-s − 12-s + 6·13-s − 15-s + 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 22-s − 24-s + 25-s + 6·26-s − 27-s − 10·29-s − 30-s + 32-s − 33-s + 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 1.85·29-s − 0.182·30-s + 0.176·32-s − 0.174·33-s + 0.342·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{330} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.877960898\)
\(L(\frac12)\) \(\approx\) \(1.877960898\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.92343107519767, −18.30012263825393, −17.32232848992984, −16.57759883357334, −15.89505621062751, −14.92404085780563, −14.14890973423609, −13.10973661998302, −12.75368980079320, −11.40228492181983, −11.03006073797187, −9.918282683247511, −8.806615184716062, −7.575732818550543, −6.269703227515569, −5.880651027382354, −4.548314311993838, −3.439076630836333, −1.627933613981861, 1.627933613981861, 3.439076630836333, 4.548314311993838, 5.880651027382354, 6.269703227515569, 7.575732818550543, 8.806615184716062, 9.918282683247511, 11.03006073797187, 11.40228492181983, 12.75368980079320, 13.10973661998302, 14.14890973423609, 14.92404085780563, 15.89505621062751, 16.57759883357334, 17.32232848992984, 18.30012263825393, 18.92343107519767

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