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Properties

Label 2-1050-105.89-c1-0-24
Degree $2$
Conductor $1050$
Sign $0.00342 + 0.999i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 1.73i·6-s + (−0.866 + 2.5i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−2.59 + 1.5i)11-s + (1.49 + 0.866i)12-s + 3.46·13-s + (−1.73 − 2i)14-s + (−0.5 + 0.866i)16-s + (−3 + 1.73i)17-s + (1.5 + 2.59i)18-s + (−3 − 1.73i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.707i·6-s + (−0.327 + 0.944i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.783 + 0.452i)11-s + (0.433 + 0.250i)12-s + 0.960·13-s + (−0.462 − 0.534i)14-s + (−0.125 + 0.216i)16-s + (−0.727 + 0.420i)17-s + (0.353 + 0.612i)18-s + (−0.688 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00342 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00342 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.00342 + 0.999i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.00342 + 0.999i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.866 - 2.5i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.73 + i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (6 + 3.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-3.46 - 6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.66iT - 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.19T + 97T^{2} \)
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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

−9.644284475153166885071287479442, −8.943172666408460464691398410332, −8.147594357469723068483311779922, −7.00875333003205074510190238527, −6.17405697917139266517252019248, −5.61605459286275017501719482348, −4.71383007952415188173498076501, −3.63324165221143756356840086905, −1.98251300528504065055767943605, 0, 1.23624692067507408870995222339, 2.60977514441919106655771403569, 3.93863113598051416985130476312, 4.79196407689664757775744195782, 6.05699670719865008516638579394, 6.68975243829269091273780813837, 7.73797270931369872945207767938, 8.346596293764472682750887984166, 9.482541967836300988794467564422

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