Properties

Degree $2$
Conductor $1050$
Sign $0.943 + 0.330i$
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 + 0.999·6-s + (1.73 + 2i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 − 0.499i)12-s − 7i·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + (0.866 + 0.499i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + 0.408·6-s + (0.654 + 0.755i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s + (0.249 − 0.144i)12-s − 1.94i·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + (0.204 + 0.117i)18-s + (0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.943 + 0.330i$
Motivic weight: \(1\)
Character: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.001861081\)
\(L(\frac12)\) \(\approx\) \(3.001861081\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-1.73 - 2i)T \)
good11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 7iT - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−10.15352144018896663460787214489, −9.007021173302293044942375593296, −8.209659347034974765477044830832, −7.60579846077249406253361375259, −6.12884177882849378549241655912, −5.45504625595180649407623300544, −4.65167387274617843381542071030, −3.37676575614118400055559400822, −2.77062533509549211143650263490, −1.35123098113178104972523446365, 1.44921933712682174067501014468, 2.63763530926095147164908056660, 4.03340493984169817848362607907, 4.45978841307789148612285048833, 5.67613177828573388864680577364, 6.86151403788306634063673009953, 7.22659008047694706034277543842, 8.146119346717198736701478128504, 9.016940062695966031595626014807, 9.846182757319326646819967675638

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