Properties

Degree $2$
Conductor $1470$
Sign $0.185 - 0.982i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.133 − 2.23i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.23 + 1.86i)10-s + (2.5 + 4.33i)11-s + (−0.866 − 0.499i)12-s i·13-s + (1 + 1.99i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−0.866 + 0.499i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.0599 − 0.998i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.389 + 0.590i)10-s + (0.753 + 1.30i)11-s + (−0.249 − 0.144i)12-s − 0.277i·13-s + (0.258 + 0.516i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.204 + 0.117i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.185 - 0.982i$
Motivic weight: \(1\)
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6317215399\)
\(L(\frac12)\) \(\approx\) \(0.6317215399\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.33 + 2.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + (-11.2 - 6.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 - 3i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.613055679221503147984652003662, −9.126456021692919872301902729346, −8.179456841830496252536288570892, −7.47320733179973674806071726507, −6.35626794314856406408041272069, −5.62939234406027827604453509770, −4.31376024040273172655938024966, −4.06546125420022607425354934876, −2.22934482189743452760916840784, −1.25792023263463398658579996005, 0.35103269051662378624537017799, 1.90965459849557283694028087086, 3.07274155075159331636336727263, 4.25250627929462847079527339121, 5.57220493575768833728711896508, 6.24123521958903892584338073862, 6.92282117787854003094393030226, 7.47103929867703972780526956605, 8.755430363384911337199096899300, 9.035295506554381182598613865668

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