Properties

Label 2-1050-35.12-c1-0-18
Degree $2$
Conductor $1050$
Sign $-0.830 + 0.556i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (0.781 − 2.52i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.31 − 2.27i)11-s + (−0.258 − 0.965i)12-s + (1.21 + 1.21i)13-s + (−0.101 + 2.64i)14-s + (0.500 − 0.866i)16-s + (−7.31 − 1.95i)17-s + (0.965 + 0.258i)18-s + (2.32 − 4.02i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (0.295 − 0.955i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.395 − 0.685i)11-s + (−0.0747 − 0.278i)12-s + (0.337 + 0.337i)13-s + (−0.0270 + 0.706i)14-s + (0.125 − 0.216i)16-s + (−1.77 − 0.475i)17-s + (0.227 + 0.0610i)18-s + (0.533 − 0.924i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.830 + 0.556i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7481869452\)
\(L(\frac12)\) \(\approx\) \(0.7481869452\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.781 + 2.52i)T \)
good11 \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.21 - 1.21i)T + 13iT^{2} \)
17 \( 1 + (7.31 + 1.95i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 - 4.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.99iT - 29T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.82 + 1.02i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (-0.545 + 0.545i)T - 43iT^{2} \)
47 \( 1 + (-1.64 - 6.12i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.28 + 2.22i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.86 - 6.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 2.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.663 + 2.47i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + (-3.53 + 13.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.78 + 4.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.99 + 7.99i)T + 83iT^{2} \)
89 \( 1 + (-0.0812 + 0.140i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.35 - 4.35i)T - 97iT^{2} \)
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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

−9.265964110442009885540769617884, −8.860189466209288228621650742714, −7.78099115571715690765996973180, −7.22678605910690465036460105401, −6.50485075955325856305383260746, −5.42506155072525731276465894343, −4.26062760383039950280577114736, −2.97998405061657319588125039693, −1.72693544441619297898427583403, −0.39607434801236351268424718123, 1.84234617764990301648166746503, 2.75516279887291328671612913181, 4.02040479242790185999501227188, 5.06359371660029204242715009379, 6.00575595200914906559200232861, 7.01955334608731691611727693042, 8.110267712440986829719676434550, 8.656639437059792237412405863338, 9.390956334161617379589847446577, 10.14368865791495371584782926487

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