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Properties

Label 2-1050-35.33-c1-0-5
Degree $2$
Conductor $1050$
Sign $0.767 - 0.641i$
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 + (2.55 − 0.703i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (0.989 + 1.71i)11-s + (0.965 − 0.258i)12-s + (−2.19 + 2.19i)13-s + (−1.33 − 2.28i)14-s + (0.500 − 0.866i)16-s + (−1.19 + 4.44i)17-s + (0.258 − 0.965i)18-s + (−2.10 + 3.65i)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 + (0.963 − 0.265i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.298 + 0.516i)11-s + (0.278 − 0.0747i)12-s + (−0.608 + 0.608i)13-s + (−0.358 − 0.609i)14-s + (0.125 − 0.216i)16-s + (−0.288 + 1.07i)17-s + (0.0610 − 0.227i)18-s + (−0.483 + 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.767 - 0.641i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9239128295\)
\(L(\frac12)\) \(\approx\) \(0.9239128295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 + 0.703i)T \)
good11 \( 1 + (-0.989 - 1.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.19 - 2.19i)T - 13iT^{2} \)
17 \( 1 + (1.19 - 4.44i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.10 - 3.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.68 - 1.52i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.94iT - 29T^{2} \)
31 \( 1 + (1.50 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.717 - 2.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + (-6.33 - 6.33i)T + 43iT^{2} \)
47 \( 1 + (5.87 - 1.57i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.95 - 11.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.10 - 3.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.63 - 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.32 - 1.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (-14.7 - 3.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.21 + 1.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.52 + 9.52i)T - 83iT^{2} \)
89 \( 1 + (3.09 - 5.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.48 + 1.48i)T + 97iT^{2} \)
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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

−10.08702449283441716309653390954, −9.456324238073349055100331696155, −8.191223098168009071325907273563, −7.79070573623912336777725510849, −6.59145326064156261819721088698, −5.69631777205677664560713731721, −4.43118394890440692769201364740, −4.09210766526681776854888697291, −2.25482903291544650842283143934, −1.43136454830919625322832142098, 0.49658883761329656973357148993, 2.21185252687015740662612610561, 3.81868775710523345878070039357, 5.03305377081427182323572400198, 5.32157377147058286228770368045, 6.51229216790641830867308850784, 7.24068860864218005285043241725, 8.159451975554489065392668063386, 8.893350825272211437404380070560, 9.704093194487787219334638012032

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