↑

Properties

Label 2-1050-7.3-c2-0-34
Degree $2$
Conductor $1050$
Sign $0.172 + 0.985i$
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 + (−6.99 + 0.325i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−6.09 + 10.5i)11-s + (−2.99 + 1.73i)12-s − 25.3i·13-s + (5.34 + 8.33i)14-s + (−2.00 − 3.46i)16-s + (24.9 + 14.3i)17-s + (2.12 − 3.67i)18-s + (−13.9 + 8.03i)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 + (−0.998 + 0.0465i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.554 + 0.959i)11-s + (−0.249 + 0.144i)12-s − 1.95i·13-s + (0.381 + 0.595i)14-s + (−0.125 − 0.216i)16-s + (1.46 + 0.846i)17-s + (0.117 − 0.204i)18-s + (−0.732 + 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.172 + 0.985i$
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: \((2,\ 1050,\ (\ :1),\ 0.172 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.302087261\)
\(L(\frac12)\) \(\approx\) \(1.302087261\)
\(L(2)\) 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 \( 1 + (6.99 - 0.325i)T \)
good11 \( 1 + (6.09 - 10.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 25.3iT - 169T^{2} \)
17 \( 1 + (-24.9 - 14.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.9 - 8.03i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.8 + 20.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 27.9T + 841T^{2} \)
31 \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (14.5 + 25.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 56.9iT - 1.68e3T^{2} \)
43 \( 1 + 7.83T + 1.84e3T^{2} \)
47 \( 1 + (-19.7 + 11.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-24.2 + 42.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (62.3 + 36.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-99.2 + 57.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (35.2 - 61.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.41T + 5.04e3T^{2} \)
73 \( 1 + (-34.2 - 19.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-27.3 - 47.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 135. iT - 6.88e3T^{2} \)
89 \( 1 + (-124. + 72.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 78.9iT - 9.40e3T^{2} \)
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

−10.02413851547967464865104589231, −8.658224174413281653860955807345, −8.118209521166108865898717119336, −7.30758575759728407773170049092, −6.07290593883121262093988894597, −5.11348310094649850762051904831, −3.84936973858866228287984596811, −3.12136218638291567739301276982, −2.18425971840452894055958213268, −0.51436342320704739688040061734, 1.01384192469592759644523032403, 2.55904089681867317032619995363, 3.58690871614798345581075874211, 4.74142605372707172184289873278, 5.98605887840407609744204523564, 6.57954632064103281358411008686, 7.41005142023696368737273087799, 8.230454626371660033789894129208, 9.084058018032403614416511277164, 9.632073623934952743558846513329

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