Properties

Degree $2$
Conductor $1470$
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 − 13-s + 15-s + 16-s − 18-s + 3·19-s + 20-s + 22-s + 7·23-s − 24-s + 25-s + 26-s + 27-s − 8·29-s − 30-s + 2·31-s − 32-s − 33-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 − 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.688·19-s + 0.223·20-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.48·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1470} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663286917\)
\(L(\frac12)\) \(\approx\) \(1.663286917\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−9.217092299643704712052668053705, −9.089567991514105168634090054667, −7.68107648255631024309043088683, −7.54284655431929603006570853838, −6.35268861081301845967524360081, −5.49912118830569381810269317967, −4.39821198685569555876220275458, −3.10630202053253186340416927410, −2.33523710039910241765913893337, −1.04195073180868306135810059408, 1.04195073180868306135810059408, 2.33523710039910241765913893337, 3.10630202053253186340416927410, 4.39821198685569555876220275458, 5.49912118830569381810269317967, 6.35268861081301845967524360081, 7.54284655431929603006570853838, 7.68107648255631024309043088683, 9.089567991514105168634090054667, 9.217092299643704712052668053705

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