↑

Properties

Label 2-1050-21.20-c1-0-23
Degree $2$
Conductor $1050$
Sign $0.890 - 0.454i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.420 − 1.68i)3-s − 4-s + (1.68 − 0.420i)6-s + (2.57 − 0.595i)7-s − i·8-s + (−2.64 + 1.41i)9-s + 2.82i·11-s + (0.420 + 1.68i)12-s + 3.36i·13-s + (0.595 + 2.57i)14-s + 16-s + 4.75·17-s + (−1.41 − 2.64i)18-s − 5.59i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.242 − 0.970i)3-s − 0.5·4-s + (0.685 − 0.171i)6-s + (0.974 − 0.224i)7-s − 0.353i·8-s + (−0.881 + 0.471i)9-s + 0.852i·11-s + (0.121 + 0.485i)12-s + 0.931i·13-s + (0.159 + 0.688i)14-s + 0.250·16-s + 1.15·17-s + (−0.333 − 0.623i)18-s − 1.28i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.528300868\)
\(L(\frac12)\) \(\approx\) \(1.528300868\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.420 + 1.68i)T \)
5 \( 1 \)
7 \( 1 + (-2.57 + 0.595i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 3.36iT - 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 + 5.59iT - 19T^{2} \)
23 \( 1 - 7.29iT - 23T^{2} \)
29 \( 1 - 0.500iT - 29T^{2} \)
31 \( 1 + 3.06iT - 31T^{2} \)
37 \( 1 + 3.32T + 37T^{2} \)
41 \( 1 - 4.33T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 7.82T + 47T^{2} \)
53 \( 1 + 8.58iT - 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 + 2.52iT - 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 9.81iT - 71T^{2} \)
73 \( 1 - 5.53iT - 73T^{2} \)
79 \( 1 + 3.29T + 79T^{2} \)
83 \( 1 - 6.97T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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

−9.761248687697726400693708015938, −8.975794750312854993553809824044, −8.014109616124754987838202776860, −7.34209407015382871092138443988, −6.91508129852195834838920589601, −5.69894874083109882426788366184, −5.05705347831554412779416909459, −3.99173095335004775470502594978, −2.30844436196626775003000930536, −1.13574900802858243086407371277, 0.924246234439513960075637866288, 2.62081572269939460713267515634, 3.58057213761551528068746400485, 4.48067204605119370696663593220, 5.49277458389787056739570401747, 5.93954007806403177303818137554, 7.73522287141994139997553852009, 8.403468123419627145487455307540, 9.084222899960760813129096490308, 10.19802848488124369739899313263

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