Properties

Degree $2$
Conductor $1050$
Sign $-0.991 + 0.126i$
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 − 7·13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-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 − 1.94·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.117 − 0.204i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.991 + 0.126i$
Motivic weight: \(1\)
Character: $\chi_{1050} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4354050197\)
\(L(\frac12)\) \(\approx\) \(0.4354050197\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.14177679857201076399853575460, −9.217847943863417654910131735616, −8.763429218826338024815924245001, −7.54655860121012267432293339449, −7.11753289541105062079139813968, −6.09874473276908627826360804635, −5.41795298118006343070344291396, −4.25137419982369498222301125032, −3.09701589158563214147548231483, −2.14682358624930695897044834236, 0.15145845653184741597426219134, 2.17770690470256555811294470287, 3.07752250320498633143776105304, 4.15593465151363062779028594369, 4.80688969827155091116748630359, 5.87742593292052611748775276339, 7.09948093994691587905881795124, 7.65515762491952398171955998303, 9.173266552524022664719450064894, 9.559796413988722056828933961406

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