Properties

Label 2-1050-35.24-c2-0-32
Degree $2$
Conductor $1050$
Sign $0.312 + 0.949i$
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  + (1.22 − 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (6.74 + 1.88i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (1.65 − 2.87i)11-s + (−1.73 − 2.99i)12-s + 19.5·13-s + (9.58 − 2.45i)14-s + (−2.00 − 3.46i)16-s + (−4.21 + 7.30i)17-s + (−3.67 − 2.12i)18-s + (−0.704 + 0.406i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (0.963 + 0.269i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.150 − 0.260i)11-s + (−0.144 − 0.249i)12-s + 1.50·13-s + (0.684 − 0.175i)14-s + (−0.125 − 0.216i)16-s + (−0.247 + 0.429i)17-s + (−0.204 − 0.117i)18-s + (−0.0370 + 0.0214i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.312 + 0.949i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.312 + 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.727754153\)
\(L(\frac12)\) \(\approx\) \(3.727754153\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-6.74 - 1.88i)T \)
good11 \( 1 + (-1.65 + 2.87i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 19.5T + 169T^{2} \)
17 \( 1 + (4.21 - 7.30i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (0.704 - 0.406i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-10.3 + 5.98i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 4.68T + 841T^{2} \)
31 \( 1 + (-27.9 - 16.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-5.21 + 3.01i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 19.6iT - 1.68e3T^{2} \)
43 \( 1 - 2.53iT - 1.84e3T^{2} \)
47 \( 1 + (9.14 + 15.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (27.6 + 15.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (64.3 + 37.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-95.1 + 54.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (81.9 + 47.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 98.0T + 5.04e3T^{2} \)
73 \( 1 + (5.55 - 9.62i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-0.500 - 0.867i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 66.8T + 6.88e3T^{2} \)
89 \( 1 + (133. - 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 132.T + 9.40e3T^{2} \)
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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

−9.502415127023295816704552381423, −8.524765362518875194361859204564, −8.111799960260600675824333741337, −6.84681557443048926736363631900, −6.11994624853078722351793704025, −5.20396853185412062875862780812, −4.15897747040621948016484435118, −3.21376413401955250000430978668, −2.00042069376476205788953625928, −1.07604546448717960894345332447, 1.35912357130700542945884261866, 2.76229205805054265069142025378, 3.90629859883362907110349203366, 4.54865247820871858449873062322, 5.47635001228138144283650662270, 6.40666286667893745794573825919, 7.41069548586652873503807299072, 8.231230012646824066790682858114, 8.861981303283448368823440930614, 9.872898815543167599346058577985

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