Properties

Label 2-1050-35.19-c2-0-37
Degree $2$
Conductor $1050$
Sign $-0.0369 + 0.999i$
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 + (−2.55 + 6.51i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−6.16 − 10.6i)11-s + (−1.73 + 2.99i)12-s + 7.26·13-s + (7.73 − 6.17i)14-s + (−2.00 + 3.46i)16-s + (−4.64 − 8.04i)17-s + (3.67 − 2.12i)18-s + (−5.26 − 3.03i)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 + (−0.365 + 0.930i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.560 − 0.970i)11-s + (−0.144 + 0.249i)12-s + 0.558·13-s + (0.552 − 0.440i)14-s + (−0.125 + 0.216i)16-s + (−0.273 − 0.473i)17-s + (0.204 − 0.117i)18-s + (−0.276 − 0.159i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0369 + 0.999i$
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: \((2,\ 1050,\ (\ :1),\ -0.0369 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7638780700\)
\(L(\frac12)\) \(\approx\) \(0.7638780700\)
\(L(2)\) 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 \( 1 + (2.55 - 6.51i)T \)
good11 \( 1 + (6.16 + 10.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.26T + 169T^{2} \)
17 \( 1 + (4.64 + 8.04i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (5.26 + 3.03i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.94 - 1.12i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 42.2T + 841T^{2} \)
31 \( 1 + (1.05 - 0.609i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.3 - 17.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 57.8iT - 1.68e3T^{2} \)
43 \( 1 + 34.0iT - 1.84e3T^{2} \)
47 \( 1 + (-28.5 + 49.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-12.5 + 7.27i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.1 - 28.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.07 + 2.93i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (42.8 - 24.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101.T + 5.04e3T^{2} \)
73 \( 1 + (-41.1 - 71.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-55.8 + 96.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 91.6T + 6.88e3T^{2} \)
89 \( 1 + (110. + 63.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 61.4T + 9.40e3T^{2} \)
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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

−9.341147476651997249164891508377, −8.842845536233335177562934419860, −8.200217796409564378545550543094, −7.16292643357584381182619667688, −6.00804874906272620800226324865, −5.28753741519562243816859457316, −3.88656330284387451931142581234, −3.01719844447765504857973131874, −2.09321413569584296310384551537, −0.29829880506918910755191296157, 1.15483854238625047862872496305, 2.30301261217089033816377501724, 3.65855127830673836919469846112, 4.71613746711048096783260414592, 6.03922976256304388033311440564, 6.69161928804715439017179222465, 7.66604043112639903486944938343, 7.946943511974090529828569380152, 9.195225168551922754439164750913, 9.723265042703197073193927280523

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