Properties

Label 2-1050-35.33-c1-0-7
Degree $2$
Conductor $1050$
Sign $-0.404 - 0.914i$
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 + (−2.46 − 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (1.92 + 3.33i)11-s + (0.965 − 0.258i)12-s + (4.42 − 4.42i)13-s + (0.295 − 2.62i)14-s + (0.500 − 0.866i)16-s + (−0.364 + 1.36i)17-s + (−0.258 + 0.965i)18-s + (−1.76 + 3.05i)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 + (−0.930 − 0.365i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.580 + 1.00i)11-s + (0.278 − 0.0747i)12-s + (1.22 − 1.22i)13-s + (0.0789 − 0.702i)14-s + (0.125 − 0.216i)16-s + (−0.0884 + 0.330i)17-s + (−0.0610 + 0.227i)18-s + (−0.404 + 0.700i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.404 - 0.914i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.404 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.062623210\)
\(L(\frac12)\) \(\approx\) \(1.062623210\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.965i)T \)
good11 \( 1 + (-1.92 - 3.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.42 + 4.42i)T - 13iT^{2} \)
17 \( 1 + (0.364 - 1.36i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.76 - 3.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 + 0.364i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.66iT - 29T^{2} \)
31 \( 1 + (4.55 - 2.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.63 - 9.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.741iT - 41T^{2} \)
43 \( 1 + (1.52 + 1.52i)T + 43iT^{2} \)
47 \( 1 + (5.07 - 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.54 - 9.51i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.84 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.857 + 0.495i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.39 + 1.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (-10.2 - 2.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.72 - 1.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.471 - 0.471i)T - 83iT^{2} \)
89 \( 1 + (-6.97 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.98 + 6.98i)T + 97iT^{2} \)
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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

−10.25900541567464696781394916753, −9.299649195302334121462224233904, −8.411712805768809708296709523461, −7.47819777500030909184888833651, −6.65054866446193663153329478687, −6.11491288799599197113705328913, −5.17494044711496861086968474127, −4.07214294398423089628965714612, −3.22728125150189199574292262614, −1.27157790254273228083276227855, 0.54794496586092242329312230589, 2.11933021737353455133866286682, 3.49205501721973240751202700714, 4.08777576707559078488820239873, 5.35202874285615596317898560599, 6.27024329477543248921655877017, 6.70289727979652968765397579632, 8.262411755913940714215300021288, 9.280304640656441654811034769775, 9.429392137057662935083849030772

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