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Properties

Label 2-1050-7.5-c2-0-24
Degree $2$
Conductor $1050$
Sign $0.464 + 0.885i$
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  + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s − 2.44i·6-s + (−5.73 + 4.01i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (8.69 + 15.0i)11-s + (−2.99 − 1.73i)12-s − 7.22i·13-s + (0.854 + 9.86i)14-s + (−2.00 + 3.46i)16-s + (2.29 − 1.32i)17-s + (−2.12 − 3.67i)18-s + (2.20 + 1.27i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s − 0.408i·6-s + (−0.819 + 0.572i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.790 + 1.36i)11-s + (−0.249 − 0.144i)12-s − 0.555i·13-s + (0.0610 + 0.704i)14-s + (−0.125 + 0.216i)16-s + (0.135 − 0.0779i)17-s + (−0.117 − 0.204i)18-s + (0.115 + 0.0668i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.464 + 0.885i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.464 + 0.885i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.630168683\)
\(L(\frac12)\) \(\approx\) \(2.630168683\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (5.73 - 4.01i)T \)
good11 \( 1 + (-8.69 - 15.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.22iT - 169T^{2} \)
17 \( 1 + (-2.29 + 1.32i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.20 - 1.27i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.0 + 34.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 47.0T + 841T^{2} \)
31 \( 1 + (-34.9 + 20.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-16.2 + 28.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 70.6iT - 1.68e3T^{2} \)
43 \( 1 + 37.3T + 1.84e3T^{2} \)
47 \( 1 + (-28.9 - 16.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (35.4 + 61.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-87.0 + 50.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (11.1 + 6.44i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-47.0 - 81.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 11.5T + 5.04e3T^{2} \)
73 \( 1 + (-19.6 + 11.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-12.0 + 20.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 111. iT - 6.88e3T^{2} \)
89 \( 1 + (110. + 63.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 7.48iT - 9.40e3T^{2} \)
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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

−9.856720617524378611576746168818, −8.863777948211425244721593301759, −8.092012837211314413193235744997, −6.79489172841379498553470463603, −6.37363019529071706590617057991, −5.02284602406525001102859354660, −4.17830126739845889298097296968, −2.99767044042449724787129600456, −2.32936883154187794415757123309, −0.906816860917272918449656944314, 1.01225230445252005517876064573, 2.99099421169217733642598360067, 3.59249584704706384583366430887, 4.53063946100009148638749557286, 5.68023344918228999550752365485, 6.56853712146838741874411897852, 7.17599008789626420242763464808, 8.301917609846793031914106644856, 8.918815276302704664765136782990, 9.675733356782724218458129628411

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