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Properties

Label 2-1050-35.24-c2-0-38
Degree $2$
Conductor $1050$
Sign $-0.270 + 0.962i$
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 + (6.98 − 0.440i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (2.76 − 4.79i)11-s + (−1.73 − 2.99i)12-s − 3.50·13-s + (8.24 − 5.47i)14-s + (−2.00 − 3.46i)16-s + (4.06 − 7.04i)17-s + (−3.67 − 2.12i)18-s + (15.3 − 8.86i)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 + (0.998 − 0.0628i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.251 − 0.435i)11-s + (−0.144 − 0.249i)12-s − 0.269·13-s + (0.588 − 0.391i)14-s + (−0.125 − 0.216i)16-s + (0.239 − 0.414i)17-s + (−0.204 − 0.117i)18-s + (0.808 − 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.270 + 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.395525200\)
\(L(\frac12)\) \(\approx\) \(3.395525200\)
\(L(2)\) 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 \( 1 + (-6.98 + 0.440i)T \)
good11 \( 1 + (-2.76 + 4.79i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.50T + 169T^{2} \)
17 \( 1 + (-4.06 + 7.04i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.3 + 8.86i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (10.3 - 5.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 8.52T + 841T^{2} \)
31 \( 1 + (6.68 + 3.85i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-24.2 + 14.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 3.14iT - 1.68e3T^{2} \)
43 \( 1 + 43.1iT - 1.84e3T^{2} \)
47 \( 1 + (9.35 + 16.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (2.44 + 1.40i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-26.1 - 15.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.0 - 23.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.70 - 3.29i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 97.7T + 5.04e3T^{2} \)
73 \( 1 + (-30.5 + 52.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-61.5 - 106. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 89.5T + 6.88e3T^{2} \)
89 \( 1 + (-102. + 59.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 65.1T + 9.40e3T^{2} \)
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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

−9.478657383089860122883483538167, −8.625346877868143462802945998474, −7.69204128946379521021198014525, −7.04225073861119515482081498274, −5.89023169129492148492821808361, −5.12348411453087967868566799962, −4.12291340036270216455184293614, −3.04071544560271140772543020563, −1.98927342138101932479785944537, −0.859644045037976033619192467890, 1.58040508197361957577634785982, 2.82175999346408411935977408621, 3.96858784703203193593195964490, 4.71341517225783155984292366447, 5.49432773108845586916899948516, 6.48461985709450207030463053038, 7.65061116946277278514725451377, 8.071669774194991317826579301702, 9.090484155244774836778748815958, 9.954268470676442419492239753892

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