Properties

Label 2-1050-35.24-c2-0-31
Degree $2$
Conductor $1050$
Sign $-0.162 + 0.986i$
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 + (−4.79 + 5.10i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (9.03 − 15.6i)11-s + (1.73 + 2.99i)12-s + 18.6·13-s + (2.26 − 9.63i)14-s + (−2.00 − 3.46i)16-s + (0.770 − 1.33i)17-s + (3.67 + 2.12i)18-s + (−29.4 + 17.0i)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 + (−0.684 + 0.728i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.821 − 1.42i)11-s + (0.144 + 0.249i)12-s + 1.43·13-s + (0.161 − 0.688i)14-s + (−0.125 − 0.216i)16-s + (0.0453 − 0.0785i)17-s + (0.204 + 0.117i)18-s + (−1.55 + 0.895i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.162 + 0.986i$
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: \((2,\ 1050,\ (\ :1),\ -0.162 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2803754778\)
\(L(\frac12)\) \(\approx\) \(0.2803754778\)
\(L(2)\) 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 \( 1 + (4.79 - 5.10i)T \)
good11 \( 1 + (-9.03 + 15.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 18.6T + 169T^{2} \)
17 \( 1 + (-0.770 + 1.33i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (29.4 - 17.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (23.3 - 13.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 16.4T + 841T^{2} \)
31 \( 1 + (24.1 + 13.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (44.7 - 25.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 37.4iT - 1.68e3T^{2} \)
43 \( 1 + 63.6iT - 1.84e3T^{2} \)
47 \( 1 + (-14.1 - 24.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (0.384 + 0.221i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (64.2 + 37.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-91.0 + 52.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (11.0 + 6.35i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 45.7T + 5.04e3T^{2} \)
73 \( 1 + (18.2 - 31.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.5 + 115. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 49.9T + 6.88e3T^{2} \)
89 \( 1 + (85.9 - 49.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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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

−9.371851438176824989745252149298, −8.582126393109172211402926672168, −8.299106115433580606593757055889, −6.67975868021744155472259757681, −6.06773846584761486780369151298, −5.64558859041787132344864071014, −4.01961395371174979270212432609, −3.28792854136197265225165748206, −1.65206639911299263009462683850, −0.11862365512831323349308496935, 1.23483859372246825077147437825, 2.26427493588113766665131886819, 3.73838434097326220890466513282, 4.43473796132225049255902339086, 6.07255753441804453442894101623, 6.75406012972576513979998053434, 7.28888031710924715190635903487, 8.475386389277005145074991784357, 9.053744008317228395315215602276, 10.11154844386374361377195181079

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