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Properties

Label 2-1050-35.3-c1-0-12
Degree $2$
Conductor $1050$
Sign $-0.0677 - 0.997i$
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 + (1.48 + 2.19i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−1.36 + 2.36i)11-s + (−0.258 + 0.965i)12-s + (3.86 − 3.86i)13-s + (0.866 + 2.49i)14-s + (0.500 + 0.866i)16-s + (−5.53 + 1.48i)17-s + (−0.965 + 0.258i)18-s + (3.09 + 5.36i)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 + (0.560 + 0.827i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.411 + 0.713i)11-s + (−0.0747 + 0.278i)12-s + (1.07 − 1.07i)13-s + (0.231 + 0.668i)14-s + (0.125 + 0.216i)16-s + (−1.34 + 0.359i)17-s + (−0.227 + 0.0610i)18-s + (0.710 + 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0677 - 0.997i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.0677 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.618236226\)
\(L(\frac12)\) \(\approx\) \(2.618236226\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 2.19i)T \)
good11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \)
17 \( 1 + (5.53 - 1.48i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.09 - 5.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0693 + 0.258i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.19iT - 29T^{2} \)
31 \( 1 + (-5.13 - 2.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.22 + 0.328i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.46iT - 41T^{2} \)
43 \( 1 + (7.20 + 7.20i)T + 43iT^{2} \)
47 \( 1 + (2.84 - 10.6i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.43 + 2.26i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.73 - 6.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.56 - 0.901i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.03 + 3.86i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + (2.07 + 7.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.9 + 8.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.00 + 4.00i)T - 83iT^{2} \)
89 \( 1 + (-0.232 - 0.401i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \)
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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

−10.35814871550398618645245068447, −9.220305591981780328737737592102, −8.338496394399179988040631808817, −7.82499043226946661433363369515, −6.51395477629947573154562509299, −5.66648985823832837117846249999, −4.98428611891913995878956381263, −4.02013013325127401869894072092, −3.00695804995083746580738795388, −1.87333421162344474701400810406, 0.975349248856731245457892670401, 2.23668431639337899672787761052, 3.40937965742944531405486553512, 4.41320876887158129627915402539, 5.23397659349038544336065249476, 6.57289796532198282823506192903, 6.83715555392056619205326218357, 8.021049406959759064931534199081, 8.730201398838950477757904716002, 9.728013611933873487490269345828

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