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Properties

Label 2-1050-21.5-c1-0-24
Degree $2$
Conductor $1050$
Sign $0.112 - 0.993i$
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 + (1.52 + 0.824i)3-s + (0.499 + 0.866i)4-s + (0.907 + 1.47i)6-s + (−2.27 − 1.35i)7-s + 0.999i·8-s + (1.64 + 2.51i)9-s + (2.84 − 1.64i)11-s + (0.0481 + 1.73i)12-s + 5.91i·13-s + (−1.29 − 2.30i)14-s + (−0.5 + 0.866i)16-s + (1.20 + 2.08i)17-s + (0.166 + 2.99i)18-s + (4.77 + 2.75i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.879 + 0.475i)3-s + (0.249 + 0.433i)4-s + (0.370 + 0.602i)6-s + (−0.859 − 0.511i)7-s + 0.353i·8-s + (0.547 + 0.836i)9-s + (0.857 − 0.495i)11-s + (0.0138 + 0.499i)12-s + 1.64i·13-s + (−0.345 − 0.617i)14-s + (−0.125 + 0.216i)16-s + (0.291 + 0.504i)17-s + (0.0392 + 0.706i)18-s + (1.09 + 0.632i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.112 - 0.993i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.961199427\)
\(L(\frac12)\) \(\approx\) \(2.961199427\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.824i)T \)
5 \( 1 \)
7 \( 1 + (2.27 + 1.35i)T \)
good11 \( 1 + (-2.84 + 1.64i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.91iT - 13T^{2} \)
17 \( 1 + (-1.20 - 2.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.77 - 2.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.62 + 3.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.80iT - 29T^{2} \)
31 \( 1 + (-4.50 + 2.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.940 - 1.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.103T + 41T^{2} \)
43 \( 1 - 1.48T + 43T^{2} \)
47 \( 1 + (-6.23 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.11 - 1.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.82 + 3.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.3 + 7.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.67 + 2.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-7.02 + 4.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.57 - 7.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.54T + 83T^{2} \)
89 \( 1 + (-5.62 + 9.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.90iT - 97T^{2} \)
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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

−9.911557533749818929096651775595, −9.284212671569731668517684818347, −8.474317005204779263026222935032, −7.54079049862301209250815923052, −6.69289980740189232139952771343, −5.97180525897280979729519168818, −4.57458863989362262562320567764, −3.87105789612522725339033209473, −3.22445053338626450641621836539, −1.79262740527511672000413350637, 1.08006415787822087069850303206, 2.59226896713619905375126993828, 3.17491301622964490046145718464, 4.15863697787011233481049279899, 5.49694920756939175025500345902, 6.25463079632535938220601591285, 7.24646685356954657007035829481, 7.937656942235186206214890488735, 9.117302010909278487620114404552, 9.661377667463089221855880330972

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