Properties

Degree $2$
Conductor $1470$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

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 − 5·11-s − 12-s − 5·13-s − 15-s + 16-s − 4·17-s + 18-s − 7·19-s + 20-s − 5·22-s + 23-s − 24-s + 25-s − 5·26-s − 27-s − 30-s − 2·31-s + 32-s + 5·33-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 − 1.50·11-s − 0.288·12-s − 1.38·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 1.60·19-s + 0.223·20-s − 1.06·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.192·27-s − 0.182·30-s − 0.359·31-s + 0.176·32-s + 0.870·33-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: $\chi_{1470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−19.64895778304321, −19.15216240993013, −18.25211184194837, −17.53199824948805, −17.12681700648412, −16.21356706195067, −15.66581006017413, −14.88567913049857, −14.36656394591657, −13.28627894407256, −12.85199811719517, −12.42345649109890, −11.30357284934732, −10.75532803744066, −10.14123210318703, −9.244940905585115, −8.114746466635844, −7.308503029617726, −6.500114962518237, −5.694459843729816, −4.898761966076431, −4.320051580466292, −2.752084984999949, −2.087656522536853, 0, 2.087656522536853, 2.752084984999949, 4.320051580466292, 4.898761966076431, 5.694459843729816, 6.500114962518237, 7.308503029617726, 8.114746466635844, 9.244940905585115, 10.14123210318703, 10.75532803744066, 11.30357284934732, 12.42345649109890, 12.85199811719517, 13.28627894407256, 14.36656394591657, 14.88567913049857, 15.66581006017413, 16.21356706195067, 17.12681700648412, 17.53199824948805, 18.25211184194837, 19.15216240993013, 19.64895778304321

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