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Properties

Label 2-1050-21.5-c1-0-15
Degree $2$
Conductor $1050$
Sign $0.444 - 0.895i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s − 1.73i·6-s + (2.5 + 0.866i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−2.59 + 1.5i)11-s + (0.866 − 1.49i)12-s + 3.46i·13-s + (1.73 + 2i)14-s + (−0.5 + 0.866i)16-s + (1.73 + 3i)17-s + (−2.59 + 1.5i)18-s + (3 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s − 0.707i·6-s + (0.944 + 0.327i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.783 + 0.452i)11-s + (0.250 − 0.433i)12-s + 0.960i·13-s + (0.462 + 0.534i)14-s + (−0.125 + 0.216i)16-s + (0.420 + 0.727i)17-s + (−0.612 + 0.353i)18-s + (0.688 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.861123549\)
\(L(\frac12)\) \(\approx\) \(1.861123549\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-1.73 - 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.31402750966063918388009112327, −8.988191970252149095437028981849, −7.964387902526364662649979652777, −7.63207537219953893704906142940, −6.59256007018404467280274912426, −5.76121921068640875020416571358, −5.06214428215013147089721184908, −4.12123810820563988124648607271, −2.52686028015882407279335422557, −1.59655640344064058892921091278, 0.75411294455275709095404475839, 2.58224891573146258590701070177, 3.62301522871064631976509813103, 4.54092089821154764966820859549, 5.46368174539132574608651646588, 5.76714669100252549729818400476, 7.30687784650404597730260467773, 8.038174583601301011790674915977, 9.175291544456497614215153975417, 10.10057168296736421552681793348

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