Properties

Degree $2$
Conductor $1050$
Sign $-0.266 + 0.963i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−2 + 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s − 13-s + (−0.499 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (1.5 − 2.59i)19-s + (2.5 + 0.866i)21-s − 0.999·22-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.408·6-s + (−0.755 + 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s − 0.277·13-s + (−0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.117 − 0.204i)18-s + (0.344 − 0.596i)19-s + (0.545 + 0.188i)21-s − 0.213·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.266 + 0.963i$
Motivic weight: \(1\)
Character: $\chi_{1050} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4198864935\)
\(L(\frac12)\) \(\approx\) \(0.4198864935\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (2.5 - 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−9.457360729719772350734879918968, −8.882483590307257550356918339787, −7.936695389193420610070054829336, −7.00022445654603292139064194290, −6.48322819521380494939101500023, −5.55049628084216852312899686130, −4.72560069828295464256433285488, −3.22455547916627464414146620371, −1.96754759774881770422516455926, −0.22795876623010563326447927617, 1.38808075254981327465629579691, 3.09905966689385924934063778873, 3.70687074262732592613556231659, 4.80677225119587617557780674671, 5.84023170335058914513951923808, 6.88513219938824136651322230346, 7.72051807282285335429615952951, 8.737203219992294665083474010118, 9.807814273144630303879117634666, 9.839674061996120406606656640868

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