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Properties

Label 2-1050-35.17-c1-0-16
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: $\chi_{1050} (157, \cdot )$
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

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

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