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Properties

Label 2-1050-7.3-c2-0-5
Degree $2$
Conductor $1050$
Sign $-0.952 - 0.303i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s − 2.44i·6-s + (−2.94 + 6.34i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−8.87 + 15.3i)11-s + (−2.99 + 1.73i)12-s + 8.10i·13-s + (9.86 − 0.879i)14-s + (−2.00 − 3.46i)16-s + (6.81 + 3.93i)17-s + (2.12 − 3.67i)18-s + (17.5 − 10.1i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s − 0.408i·6-s + (−0.421 + 0.907i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.806 + 1.39i)11-s + (−0.249 + 0.144i)12-s + 0.623i·13-s + (0.704 − 0.0628i)14-s + (−0.125 − 0.216i)16-s + (0.401 + 0.231i)17-s + (0.117 − 0.204i)18-s + (0.926 − 0.534i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.952 - 0.303i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.952 - 0.303i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4806836371\)
\(L(\frac12)\) \(\approx\) \(0.4806836371\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2.94 - 6.34i)T \)
good11 \( 1 + (8.87 - 15.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 8.10iT - 169T^{2} \)
17 \( 1 + (-6.81 - 3.93i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-17.5 + 10.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (16.2 + 28.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 25.0T + 841T^{2} \)
31 \( 1 + (46.0 + 26.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (21.7 + 37.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 70.3iT - 1.68e3T^{2} \)
43 \( 1 - 2.43T + 1.84e3T^{2} \)
47 \( 1 + (-13.3 + 7.69i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-25.4 + 44.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-22.9 - 13.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (28.7 - 16.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.9 + 86.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 97.0T + 5.04e3T^{2} \)
73 \( 1 + (46.7 + 27.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-37.3 - 64.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 85.2iT - 6.88e3T^{2} \)
89 \( 1 + (115. - 66.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 49.9iT - 9.40e3T^{2} \)
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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

−9.777544219385894341676689459662, −9.527663888296676813698909501487, −8.651811553095191221435652262323, −7.75031674816749091426793415430, −6.99810788200585273408201801530, −5.66883343064099734145053454255, −4.72903052254439065551064871629, −3.72281219866543927004819731377, −2.57317902991794281105626217371, −1.92229738159327407627687638758, 0.15471428119657618314729528479, 1.38034133476129368541308579182, 3.13408947449264557521308337633, 3.75235373394224225997345303279, 5.40493511204102953304274981732, 5.84680595840627078931122363323, 7.23205266570556153114434577117, 7.53917748153783818980275629814, 8.392920160530435485294985491152, 9.212633044201927018616672668979

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