Properties

Label 2-1050-3.2-c2-0-35
Degree $2$
Conductor $1050$
Sign $0.982 - 0.184i$
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 + (−0.554 − 2.94i)3-s − 2.00·4-s + (4.16 − 0.783i)6-s + 2.64·7-s − 2.82i·8-s + (−8.38 + 3.26i)9-s + 2.58i·11-s + (1.10 + 5.89i)12-s − 0.180·13-s + 3.74i·14-s + 4.00·16-s + 7.25i·17-s + (−4.62 − 11.8i)18-s + 21.4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.184 − 0.982i)3-s − 0.500·4-s + (0.694 − 0.130i)6-s + 0.377·7-s − 0.353i·8-s + (−0.931 + 0.363i)9-s + 0.234i·11-s + (0.0923 + 0.491i)12-s − 0.0138·13-s + 0.267i·14-s + 0.250·16-s + 0.426i·17-s + (−0.256 − 0.658i)18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.982 - 0.184i$
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: \((2,\ 1050,\ (\ :1),\ 0.982 - 0.184i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.632977032\)
\(L(\frac12)\) \(\approx\) \(1.632977032\)
\(L(2)\) 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 \( 1 + (0.554 + 2.94i)T \)
5 \( 1 \)
7 \( 1 - 2.64T \)
good11 \( 1 - 2.58iT - 121T^{2} \)
13 \( 1 + 0.180T + 169T^{2} \)
17 \( 1 - 7.25iT - 289T^{2} \)
19 \( 1 - 21.4T + 361T^{2} \)
23 \( 1 + 11.8iT - 529T^{2} \)
29 \( 1 - 29.5iT - 841T^{2} \)
31 \( 1 + 49.5T + 961T^{2} \)
37 \( 1 - 43.2T + 1.36e3T^{2} \)
41 \( 1 + 20.7iT - 1.68e3T^{2} \)
43 \( 1 - 74.2T + 1.84e3T^{2} \)
47 \( 1 - 7.05iT - 2.20e3T^{2} \)
53 \( 1 - 55.0iT - 2.80e3T^{2} \)
59 \( 1 + 100. iT - 3.48e3T^{2} \)
61 \( 1 - 8.44T + 3.72e3T^{2} \)
67 \( 1 - 85.7T + 4.48e3T^{2} \)
71 \( 1 + 97.8iT - 5.04e3T^{2} \)
73 \( 1 - 17.8T + 5.32e3T^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 - 3.08iT - 6.88e3T^{2} \)
89 \( 1 + 80.1iT - 7.92e3T^{2} \)
97 \( 1 - 105.T + 9.40e3T^{2} \)
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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

−9.446983726685847371149949647334, −8.760163446203762861549830196885, −7.74940352778249106723168573101, −7.38780920133407521777624886052, −6.41602160236650867657561180202, −5.62434456617978702633470798213, −4.82411330254773042230890448802, −3.48192288145639748459072464966, −2.09151034684958903366157864822, −0.823507998126972648529275712365, 0.78402015853705361926283933020, 2.41227366467757849123642939306, 3.46815767047823313749337336176, 4.28333052305307972181872005128, 5.25394812216225873213201842887, 5.89256607093019190019719593043, 7.36056845974139384378655162278, 8.253560336639344429791384084935, 9.280621260059162487108087083526, 9.632250947156805454052948634189

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