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 + 4·11-s + 12-s + 2·13-s − 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 4·22-s − 8·23-s + 24-s + 25-s + 2·26-s + 27-s + 6·29-s − 30-s + 8·31-s + 32-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 + 1.20·11-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 + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-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: $\chi_{1470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.304147990\)
\(L(\frac12)\) \(\approx\) \(3.304147990\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.97312831300784, −19.73882245663456, −18.81273675681152, −18.12852659560011, −17.28638938742541, −16.48478761604381, −15.70231642191006, −15.41365339647179, −14.45339483360051, −13.82832785034823, −13.56842591839801, −12.29249060032714, −11.97730337724744, −11.25452300510649, −10.23308566153554, −9.546464423403827, −8.488553301677911, −8.034244690198991, −6.843845188271373, −6.416271536650673, −5.228729059360876, −4.158054565895294, −3.681157481568817, −2.575249844729578, −1.290928048762499, 1.290928048762499, 2.575249844729578, 3.681157481568817, 4.158054565895294, 5.228729059360876, 6.416271536650673, 6.843845188271373, 8.034244690198991, 8.488553301677911, 9.546464423403827, 10.23308566153554, 11.25452300510649, 11.97730337724744, 12.29249060032714, 13.56842591839801, 13.82832785034823, 14.45339483360051, 15.41365339647179, 15.70231642191006, 16.48478761604381, 17.28638938742541, 18.12852659560011, 18.81273675681152, 19.73882245663456, 19.97312831300784

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