Properties

Label 2-1050-7.3-c2-0-18
Degree $2$
Conductor $1050$
Sign $-0.988 - 0.154i$
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 + (−1.94 + 6.72i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (0.263 − 0.455i)11-s + (−2.99 + 1.73i)12-s + 4.22i·13-s + (−9.61 + 2.37i)14-s + (−2.00 − 3.46i)16-s + (28.8 + 16.6i)17-s + (−2.12 + 3.67i)18-s + (2.75 − 1.58i)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 + (−0.277 + 0.960i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.0239 − 0.0414i)11-s + (−0.249 + 0.144i)12-s + 0.324i·13-s + (−0.686 + 0.169i)14-s + (−0.125 − 0.216i)16-s + (1.69 + 0.980i)17-s + (−0.117 + 0.204i)18-s + (0.144 − 0.0836i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.988 - 0.154i$
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: \((2,\ 1050,\ (\ :1),\ -0.988 - 0.154i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.116233235\)
\(L(\frac12)\) \(\approx\) \(2.116233235\)
\(L(2)\) 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 \( 1 + (1.94 - 6.72i)T \)
good11 \( 1 + (-0.263 + 0.455i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 4.22iT - 169T^{2} \)
17 \( 1 + (-28.8 - 16.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.75 + 1.58i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-5.52 - 9.57i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 56.1T + 841T^{2} \)
31 \( 1 + (1.63 + 0.945i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (4.87 + 8.44i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 4.07iT - 1.68e3T^{2} \)
43 \( 1 + 46.3T + 1.84e3T^{2} \)
47 \( 1 + (54.7 - 31.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (23.2 - 40.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (43.7 + 25.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-89.4 + 51.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-4.36 + 7.56i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 29.0T + 5.04e3T^{2} \)
73 \( 1 + (14.4 + 8.36i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-66.1 - 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 12.4iT - 6.88e3T^{2} \)
89 \( 1 + (59.1 - 34.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 149. iT - 9.40e3T^{2} \)
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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

−9.696068508188088121959737632978, −9.353600858252917744805640751149, −8.290042996563791841884361565629, −7.78185697354677584169088278564, −6.68293809190708240075952364148, −5.73122132230142350046128425324, −5.14221294686432238096712350326, −3.80932601286601190134134815453, −3.15392803134134340679313763605, −1.78131794164975734450631668008, 0.53604637427662870687220323227, 1.69495637192620017006506529833, 3.11829387268171028861266178933, 3.62831419079796483872322257846, 4.83809137070022435486397451007, 5.76104487265241621477724221595, 6.97123935283644895362542441040, 7.58214648780504862113424075514, 8.518374767297165515081683889283, 9.728369812423171173496292388200

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