Properties

Label 2-1050-7.3-c2-0-7
Degree $2$
Conductor $1050$
Sign $-0.520 - 0.853i$
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 + (6.38 − 2.86i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−9.98 + 17.2i)11-s + (−2.99 + 1.73i)12-s − 3.49i·13-s + (−8.02 − 5.80i)14-s + (−2.00 − 3.46i)16-s + (−15.7 − 9.12i)17-s + (2.12 − 3.67i)18-s + (−21.3 + 12.3i)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 + (0.912 − 0.408i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.907 + 1.57i)11-s + (−0.249 + 0.144i)12-s − 0.269i·13-s + (−0.572 − 0.414i)14-s + (−0.125 − 0.216i)16-s + (−0.929 − 0.536i)17-s + (0.117 − 0.204i)18-s + (−1.12 + 0.647i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.520 - 0.853i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7138345842\)
\(L(\frac12)\) \(\approx\) \(0.7138345842\)
\(L(2)\) 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 \( 1 + (-6.38 + 2.86i)T \)
good11 \( 1 + (9.98 - 17.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.49iT - 169T^{2} \)
17 \( 1 + (15.7 + 9.12i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (21.3 - 12.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.5 - 21.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 53.1T + 841T^{2} \)
31 \( 1 + (-26.0 - 15.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (23.3 + 40.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 31.5iT - 1.68e3T^{2} \)
43 \( 1 + 64.4T + 1.84e3T^{2} \)
47 \( 1 + (-24.3 + 14.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (32.4 - 56.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-86.7 - 50.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.94 + 4.01i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.13 + 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 107.T + 5.04e3T^{2} \)
73 \( 1 + (44.7 + 25.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.9 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 0.417iT - 6.88e3T^{2} \)
89 \( 1 + (96.3 - 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 74.2iT - 9.40e3T^{2} \)
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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

−10.14220226969386093250939698066, −9.207695462697528528762420483698, −8.483264371984271612879834835323, −7.55005738808120852405539398559, −7.13299098086113807400289239994, −5.37228019663819243086269216979, −4.58987826585938900911824796632, −3.78701490285282346567488929110, −2.39424717330093148122486938913, −1.70980523777215170586983426079, 0.21498287632065604004210717235, 1.79106334731846615941163192384, 2.86451425987261879624135929775, 4.28681227066640265336044490038, 5.22996407577117797790734299036, 6.18008377239939312370006667471, 6.93724698794978331451449951217, 8.166206956547328630548696404784, 8.395670333009227196368730976731, 9.008816671144870323689516008990

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