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Properties

Label 2-1470-35.4-c1-0-28
Degree $2$
Conductor $1470$
Sign $0.897 + 0.441i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (1.86 + 1.23i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (2.23 + 0.133i)10-s + (2.5 − 4.33i)11-s + (0.866 − 0.499i)12-s − i·13-s + (1 + 2i)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 + (0.834 + 0.550i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.705 + 0.0423i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.481321003\)
\(L(\frac12)\) \(\approx\) \(3.481321003\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.23i)T \)
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

−9.364063329898140248048015800774, −8.960897414176560162148578437836, −7.87828639610174524810058006010, −6.70961437116929632648492208728, −6.17206546255510874341110078811, −5.25479737088151368307019615635, −4.25231144002982700438526931217, −3.17778809265516785632724721791, −2.64420450594449491502242355548, −1.25883331463203513692296584288, 1.55372784150797293839212904482, 2.31797635134604043994652416418, 3.73455847310880372740679756228, 4.48580288724954478520821387002, 5.46874296862761588396016630138, 6.29448405768742356315335945177, 7.04187669664867675558769859054, 7.88030347351406621172429427169, 8.723514211821797716439088275249, 9.569787151205637830956441698342

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