Properties

Label 2-1470-35.4-c1-0-17
Degree $2$
Conductor $1470$
Sign $0.943 + 0.330i$
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 + (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 + (−1 + 1.73i)11-s + (−0.866 + 0.499i)12-s − 6i·13-s + (1 − 1.99i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (−0.866 − 0.499i)18-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 + (−0.301 + 0.522i)11-s + (−0.249 + 0.144i)12-s − 1.66i·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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.943 + 0.330i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8708326706\)
\(L(\frac12)\) \(\approx\) \(0.8708326706\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
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

−9.782904012736273616980722425805, −8.442323891463334527925734846472, −7.73406528286800490707871481343, −7.12492774752850324512130069040, −6.29965613699331467012539237489, −5.61965115653977744326222805700, −4.61656698214323637008634901469, −3.12026566243053883354757650983, −2.22808566479561012462342435614, −0.57888197889722498363021273807, 0.974165179803274292867103777158, 2.10481797138180298725424723454, 3.59036905987497069791817264788, 4.51442523509593169177359648808, 5.28256237295560185596622820414, 6.37391262335333477219694472631, 7.11042891740407071941076229614, 8.289863268740059580271871386859, 8.857237781694999914596945519629, 9.464231018467644353391284848187

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