Properties

Label 2-1050-7.3-c2-0-48
Degree $2$
Conductor $1050$
Sign $0.355 + 0.934i$
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 + (−6.80 + 1.64i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (8.87 − 15.3i)11-s + (−2.99 + 1.73i)12-s + 2.00i·13-s + (−6.82 − 7.17i)14-s + (−2.00 − 3.46i)16-s + (−18.3 − 10.6i)17-s + (−2.12 + 3.67i)18-s + (−5.70 + 3.29i)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 + (−0.972 + 0.234i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.807 − 1.39i)11-s + (−0.249 + 0.144i)12-s + 0.154i·13-s + (−0.487 − 0.512i)14-s + (−0.125 − 0.216i)16-s + (−1.08 − 0.623i)17-s + (−0.117 + 0.204i)18-s + (−0.300 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.355 + 0.934i$
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: \((2,\ 1050,\ (\ :1),\ 0.355 + 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.037741696\)
\(L(\frac12)\) \(\approx\) \(1.037741696\)
\(L(2)\) 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 \( 1 + (6.80 - 1.64i)T \)
good11 \( 1 + (-8.87 + 15.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 2.00iT - 169T^{2} \)
17 \( 1 + (18.3 + 10.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.70 - 3.29i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (19.7 + 34.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 25.1T + 841T^{2} \)
31 \( 1 + (-4.76 - 2.75i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.87 + 10.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 16.8iT - 1.68e3T^{2} \)
43 \( 1 + 27.7T + 1.84e3T^{2} \)
47 \( 1 + (39.5 - 22.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-49.2 + 85.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-10.9 - 6.34i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-82.0 + 47.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-4.17 + 7.23i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 71.8T + 5.04e3T^{2} \)
73 \( 1 + (-0.697 - 0.402i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (58.1 + 100. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 50.1iT - 6.88e3T^{2} \)
89 \( 1 + (-74.9 + 43.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 165. iT - 9.40e3T^{2} \)
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

−9.288198834847532631490558500901, −8.737376080398600473885933158027, −8.051178603430188573240342087299, −6.66573559282772576601204775039, −6.42542483460568924131406077851, −5.30172636291218815457929545282, −4.12646591381484179317445793693, −3.44317980131608036535598543208, −2.36697987803239533572608770056, −0.25181347492731309059846632278, 1.53307746954564189439928103000, 2.43975748692165006467841433801, 3.73444747231755251894263013620, 4.19763105103222008209402297913, 5.57107652676309572021276410194, 6.64009044621648993409052149772, 7.17051513661507644295312031008, 8.358240893674324806784026066685, 9.382427639280320256888181915579, 9.726915623456090471208734320412

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