↑

Properties

Label 2-1050-7.5-c2-0-47
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

Related objects

Downloads

Learn more

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: $\chi_{1050} (901, \cdot )$
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 \( 1 + 49.9iT - 9.40e3T^{2} \)
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

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

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