Properties

Label 2-1050-35.19-c2-0-20
Degree $2$
Conductor $1050$
Sign $0.831 - 0.555i$
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  + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s − 2.44i·6-s + (6.77 − 1.74i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−3 − 5.19i)11-s + (−1.73 + 2.99i)12-s − 21.3·13-s + (−9.53 − 2.65i)14-s + (−2.00 + 3.46i)16-s + (4.47 + 7.75i)17-s + (3.67 − 2.12i)18-s + (6.25 + 3.61i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s − 0.408i·6-s + (0.968 − 0.248i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.272 − 0.472i)11-s + (−0.144 + 0.249i)12-s − 1.64·13-s + (−0.681 − 0.189i)14-s + (−0.125 + 0.216i)16-s + (0.263 + 0.456i)17-s + (0.204 − 0.117i)18-s + (0.329 + 0.190i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.588804230\)
\(L(\frac12)\) \(\approx\) \(1.588804230\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-6.77 + 1.74i)T \)
good11 \( 1 + (3 + 5.19i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.3T + 169T^{2} \)
17 \( 1 + (-4.47 - 7.75i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.25 - 3.61i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-32.4 - 18.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (-38.2 + 22.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (24.2 + 13.9i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 1.48iT - 1.84e3T^{2} \)
47 \( 1 + (21.5 - 37.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (74.0 - 42.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-35.6 + 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.02 + 0.594i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (3.80 - 2.19i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (-39.4 - 68.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.1 + 85.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + (-18 - 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 10.9T + 9.40e3T^{2} \)
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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

−9.795199090720382398816405936809, −9.119278719831802937221590370015, −8.009982992118415638886761708635, −7.78225414062527027102533356222, −6.60822613870403790946312828938, −5.19873582557427132843906372560, −4.60485434481237927420600197370, −3.30410531069305064906965322222, −2.38634561439140460998650701089, −1.02784114524428917915344318758, 0.71331569752112497765568035548, 2.05255201060356710295083610828, 2.89600400218659627488735502906, 4.85401286299366214340289296645, 5.11557096138511869118585182003, 6.68341362253227700226773026254, 7.17540768076072136876608530722, 8.026252323217177589718588459769, 8.631498158723159049682440164235, 9.558210882295394819062528533467

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