Properties

Degree $2$
Conductor $1050$
Sign $0.758 - 0.652i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

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 + (−2.59 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.5 + 4.33i)11-s + (−0.866 + 0.499i)12-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + (0.866 + 0.499i)18-s + (4 + 6.92i)19-s + (2.5 + 0.866i)21-s + 5i·22-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 + (−0.981 + 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.753 + 1.30i)11-s + (−0.249 + 0.144i)12-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + (0.204 + 0.117i)18-s + (0.917 + 1.58i)19-s + (0.545 + 0.188i)21-s + 1.06i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.758 - 0.652i$
Motivic weight: \(1\)
Character: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.399032680\)
\(L(\frac12)\) \(\approx\) \(1.399032680\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 0.5i)T \)
good11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7iT - 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.10269567857564702414554259797, −9.597991056064443436478238890433, −8.205218598157456482141996757218, −7.29317497113806756899872891544, −6.52152700919763019293472512045, −5.62078783408022173939881494122, −4.92436881377592436431929597252, −3.71571472285261648147546175966, −2.73979647377619173140643440214, −1.42711203661123366459042099296, 0.57234456117050874678171243445, 2.95418766513610313660702410090, 3.39961154514182652700317008574, 4.83927145360409769722314477828, 5.45291232493097690878108713892, 6.33117746254756987682754920623, 7.09203633572202663977570051210, 7.988824873398925967319477934472, 9.106938615991996137213939911630, 9.787686932739782184045027062756

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