↑

Properties

Label 2-1050-7.5-c2-0-49
Degree $2$
Conductor $1050$
Sign $-0.874 - 0.484i$
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 + (2.59 − 6.50i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−5.13 − 8.89i)11-s + (−2.99 − 1.73i)12-s + 7.02i·13-s + (−6.12 − 7.77i)14-s + (−2.00 + 3.46i)16-s + (−27.4 + 15.8i)17-s + (−2.12 − 3.67i)18-s + (−26.9 − 15.5i)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 + (0.370 − 0.928i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.466 − 0.808i)11-s + (−0.249 − 0.144i)12-s + 0.540i·13-s + (−0.437 − 0.555i)14-s + (−0.125 + 0.216i)16-s + (−1.61 + 0.933i)17-s + (−0.117 − 0.204i)18-s + (−1.41 − 0.818i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.874 - 0.484i$
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: \((2,\ 1050,\ (\ :1),\ -0.874 - 0.484i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.264690976\)
\(L(\frac12)\) \(\approx\) \(1.264690976\)
\(L(2)\) 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 \( 1 + (-2.59 + 6.50i)T \)
good11 \( 1 + (5.13 + 8.89i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.02iT - 169T^{2} \)
17 \( 1 + (27.4 - 15.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (11.8 - 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 9.19T + 841T^{2} \)
31 \( 1 + (-17.4 + 10.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-24.0 + 41.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 65.1iT - 1.68e3T^{2} \)
43 \( 1 - 3.03T + 1.84e3T^{2} \)
47 \( 1 + (-53.6 - 30.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.690 - 1.19i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (95.1 - 54.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.3 + 19.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.95 - 13.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.3T + 5.04e3T^{2} \)
73 \( 1 + (62.6 - 36.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.2 - 92.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 49.4iT - 6.88e3T^{2} \)
89 \( 1 + (142. + 82.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 49.4iT - 9.40e3T^{2} \)
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

−9.104084940100736070425364701614, −8.595812588409614395935576450626, −7.61829096350814101714663312200, −6.67845417765755670138397034326, −5.82909305143786647975681276149, −4.36312128141918983745855220138, −4.04317750507130605099510090742, −2.65216427912262814805650034257, −1.74562007360242427525787292095, −0.29466097800851788997578981268, 2.12960172030448199736593771689, 2.88442563632156292043144741348, 4.46871019934043396538834580465, 4.74841406483640435012816435138, 6.01690741730933395334944530785, 6.73565696906261616090449332447, 7.915773596573527498664400404507, 8.395037466621927938011436054627, 9.167206855149692202016023046439, 10.07256386025325246833232690748

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