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Properties

Label 2-1050-35.19-c2-0-47
Degree $2$
Conductor $1050$
Sign $-0.0369 + 0.999i$
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 + (2.55 − 6.51i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−5.79 − 10.0i)11-s + (−1.73 + 2.99i)12-s − 7.86·13-s + (7.73 − 6.17i)14-s + (−2.00 + 3.46i)16-s + (−13.8 − 23.9i)17-s + (−3.67 + 2.12i)18-s + (−27.2 − 15.7i)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 + (0.365 − 0.930i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.526 − 0.912i)11-s + (−0.144 + 0.249i)12-s − 0.604·13-s + (0.552 − 0.440i)14-s + (−0.125 + 0.216i)16-s + (−0.811 − 1.40i)17-s + (−0.204 + 0.117i)18-s + (−1.43 − 0.826i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0369 + 0.999i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.0369 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.353098834\)
\(L(\frac12)\) \(\approx\) \(1.353098834\)
\(L(2)\) 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 \( 1 + (-2.55 + 6.51i)T \)
good11 \( 1 + (5.79 + 10.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.86T + 169T^{2} \)
17 \( 1 + (13.8 + 23.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (27.2 + 15.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (15.7 + 9.07i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 2.30T + 841T^{2} \)
31 \( 1 + (4.55 - 2.63i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.72 - 0.993i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.1iT - 1.68e3T^{2} \)
43 \( 1 - 49.8iT - 1.84e3T^{2} \)
47 \( 1 + (38.3 - 66.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-49.4 + 28.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-60.9 + 35.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (58.5 + 33.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (85.0 - 49.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 34.2T + 5.04e3T^{2} \)
73 \( 1 + (-9.74 - 16.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-45.2 + 78.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 133.T + 6.88e3T^{2} \)
89 \( 1 + (-9.58 - 5.53i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 72.3T + 9.40e3T^{2} \)
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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

−9.470371233508950707215856742095, −8.545648687578019698249081269524, −7.81765741372179640782995308008, −6.95697427981982722476762435126, −6.09522433630785052703302081149, −4.75861944615035546180158106975, −4.54518842807695375914783671864, −3.26348348446652831261402896553, −2.33193546568121542983666442178, −0.28585740726936283840407956978, 2.03072630964094498075673913086, 2.13418268229539601977310395339, 3.71745365573894430755051992711, 4.63813883580260039145052942946, 5.66569179845073092823136633172, 6.38797237154151846323776594906, 7.41677885210206019667740668998, 8.312320609448025844519435344595, 8.989437483022709547745822816164, 10.16620133418588966985922633497

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