Properties

Label 2-1050-35.33-c1-0-3
Degree $2$
Conductor $1050$
Sign $0.631 - 0.775i$
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  + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (0.258 + 2.63i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−2.54 − 4.40i)11-s + (0.965 − 0.258i)12-s + (2.02 − 2.02i)13-s + (2.47 − 0.931i)14-s + (0.500 − 0.866i)16-s + (−1.79 + 6.71i)17-s + (0.258 − 0.965i)18-s + (1.79 − 3.11i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (0.0978 + 0.995i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.767 − 1.32i)11-s + (0.278 − 0.0747i)12-s + (0.561 − 0.561i)13-s + (0.661 − 0.248i)14-s + (0.125 − 0.216i)16-s + (−0.436 + 1.62i)17-s + (0.0610 − 0.227i)18-s + (0.412 − 0.714i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7318739393\)
\(L(\frac12)\) \(\approx\) \(0.7318739393\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.258 - 2.63i)T \)
good11 \( 1 + (2.54 + 4.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.02 + 2.02i)T - 13iT^{2} \)
17 \( 1 + (1.79 - 6.71i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.79 + 3.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.71 - 1.79i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.81iT - 29T^{2} \)
31 \( 1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.01 - 7.50i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.13iT - 41T^{2} \)
43 \( 1 + (-7.84 - 7.84i)T + 43iT^{2} \)
47 \( 1 + (1.79 - 0.482i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.235 + 0.879i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.84 - 0.762i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.86T + 71T^{2} \)
73 \( 1 + (13.0 + 3.50i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.4 - 6.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.99 - 1.99i)T - 83iT^{2} \)
89 \( 1 + (3.82 - 6.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 - 13.4i)T + 97iT^{2} \)
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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

−10.26990084634741660603696780821, −9.215259944622710356252294475185, −8.340106904396414581707731962248, −7.955741725420884448830891955252, −6.32882652445335590694381667114, −5.79875269845061869941584889262, −4.89801806333817772416660890121, −3.58385310299800019924715248027, −2.63958205585200026737820215830, −1.29390611527076235331765320417, 0.40736101717178560589706915186, 2.12350804937858081007753595971, 4.00624902509382598892950267251, 4.56195567468948246170709089944, 5.56059563262751104881624734173, 6.50125070366912868708564611826, 7.40149828681895385985207228551, 7.74026463958317489638439372657, 9.054332879663560469097936335110, 9.922547425910921518239833681200

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