Properties

Label 2-1050-3.2-c2-0-67
Degree $2$
Conductor $1050$
Sign $0.569 + 0.821i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (2.46 − 1.70i)3-s − 2.00·4-s + (2.41 + 3.48i)6-s + 2.64·7-s − 2.82i·8-s + (3.15 − 8.42i)9-s − 2.66i·11-s + (−4.93 + 3.41i)12-s − 5.56·13-s + 3.74i·14-s + 4.00·16-s − 18.5i·17-s + (11.9 + 4.46i)18-s − 0.634·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.821 − 0.569i)3-s − 0.500·4-s + (0.402 + 0.581i)6-s + 0.377·7-s − 0.353i·8-s + (0.350 − 0.936i)9-s − 0.242i·11-s + (−0.410 + 0.284i)12-s − 0.427·13-s + 0.267i·14-s + 0.250·16-s − 1.09i·17-s + (0.662 + 0.248i)18-s − 0.0334·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.172656004\)
\(L(\frac12)\) \(\approx\) \(2.172656004\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-2.46 + 1.70i)T \)
5 \( 1 \)
7 \( 1 - 2.64T \)
good11 \( 1 + 2.66iT - 121T^{2} \)
13 \( 1 + 5.56T + 169T^{2} \)
17 \( 1 + 18.5iT - 289T^{2} \)
19 \( 1 + 0.634T + 361T^{2} \)
23 \( 1 - 15.6iT - 529T^{2} \)
29 \( 1 + 43.3iT - 841T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 - 35.0T + 1.36e3T^{2} \)
41 \( 1 + 15.6iT - 1.68e3T^{2} \)
43 \( 1 + 64.4T + 1.84e3T^{2} \)
47 \( 1 + 52.5iT - 2.20e3T^{2} \)
53 \( 1 + 51.6iT - 2.80e3T^{2} \)
59 \( 1 - 32.9iT - 3.48e3T^{2} \)
61 \( 1 + 104.T + 3.72e3T^{2} \)
67 \( 1 - 113.T + 4.48e3T^{2} \)
71 \( 1 - 36.8iT - 5.04e3T^{2} \)
73 \( 1 - 144.T + 5.32e3T^{2} \)
79 \( 1 + 57.9T + 6.24e3T^{2} \)
83 \( 1 + 45.0iT - 6.88e3T^{2} \)
89 \( 1 + 80.9iT - 7.92e3T^{2} \)
97 \( 1 - 96.2T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.477349599239005501054982876648, −8.525432183017247443367565745015, −7.904427062357952587163648496735, −7.20464368695726807743165450333, −6.41928917226572444525215166894, −5.36856664617443002031608141597, −4.35536018461000049620517044729, −3.24200934531030027214328859158, −2.10056124566850806305685245734, −0.61312679515156746603601382570, 1.47797811211788380873984809914, 2.51967297366802844004819358630, 3.49352084998109541030525644570, 4.45229248893827493109825928485, 5.11934503358583359677500155171, 6.48196656851558372048383520018, 7.72689234924923464272803224135, 8.333629833592557722512144858236, 9.140489961621777316996736724922, 9.870645084446965077485460058956

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