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 − 2·13-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s + 20-s − 22-s + 24-s + 25-s − 2·26-s + 27-s − 2·29-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 − 0.554·13-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 − 0.392·26-s + 0.192·27-s − 0.371·29-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

\(L(1)\) \(\approx\) \(2.418868993\)
\(L(\frac12)\) \(\approx\) \(2.418868993\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.15598378052134, −18.42129051313956, −17.31402187531808, −16.61065168968586, −15.59949094737988, −14.83522003568331, −14.22317375853940, −13.32608892761720, −12.70592134214731, −11.76607054701325, −10.60265529227596, −9.851457918454412, −8.737698669771691, −7.696193047293605, −6.692959703534273, −5.565079261822061, −4.487035149034387, −3.197985806998131, −2.001502616451070, 2.001502616451070, 3.197985806998131, 4.487035149034387, 5.565079261822061, 6.692959703534273, 7.696193047293605, 8.737698669771691, 9.851457918454412, 10.60265529227596, 11.76607054701325, 12.70592134214731, 13.32608892761720, 14.22317375853940, 14.83522003568331, 15.59949094737988, 16.61065168968586, 17.31402187531808, 18.42129051313956, 19.15598378052134

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