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Properties

Label 2-1050-35.33-c1-0-14
Degree $2$
Conductor $1050$
Sign $-0.542 + 0.839i$
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 + (−2.22 + 1.42i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.230 − 0.399i)11-s + (0.965 − 0.258i)12-s + (4.00 − 4.00i)13-s + (1.95 + 1.78i)14-s + (0.500 − 0.866i)16-s + (−0.424 + 1.58i)17-s + (0.258 − 0.965i)18-s + (−2.91 + 5.04i)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 + (−0.841 + 0.539i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.0695 − 0.120i)11-s + (0.278 − 0.0747i)12-s + (1.11 − 1.11i)13-s + (0.522 + 0.476i)14-s + (0.125 − 0.216i)16-s + (−0.102 + 0.384i)17-s + (0.0610 − 0.227i)18-s + (−0.668 + 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.542 + 0.839i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.542 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7904207958\)
\(L(\frac12)\) \(\approx\) \(0.7904207958\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 1.42i)T \)
good11 \( 1 + (0.230 + 0.399i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.00 + 4.00i)T - 13iT^{2} \)
17 \( 1 + (0.424 - 1.58i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.91 - 5.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.26 + 1.14i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (-0.0280 + 0.0162i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.08 + 7.78i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (4.75 + 4.75i)T + 43iT^{2} \)
47 \( 1 + (-8.73 + 2.33i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.710 - 2.65i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.958 - 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.7 + 6.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.68 + 0.986i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (-3.91 - 1.05i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.38 + 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.08 - 1.08i)T - 83iT^{2} \)
89 \( 1 + (5.71 - 9.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.51 + 2.51i)T + 97iT^{2} \)
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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

−9.809541860527823964849205739130, −8.830468794089655019630548676853, −8.225217500745336374629155537099, −7.08928749174879214925371115313, −5.96170159381261515935732384997, −5.57248988464409401772012681830, −4.07852589871079223648730194117, −3.26584510856312597401510890034, −2.01131060691889314141088970643, −0.47309619723127227669725810601, 1.18209355435916007022547478454, 3.12749569817368667145698236743, 4.29172795695304226221397261384, 5.01053173770474286110972149276, 6.34445592962983929337057320996, 6.61204527274708294250187017207, 7.45091726584068751758521259930, 8.724466706482755513034621171443, 9.237063975755461036629283723262, 10.10900775187935469211400613945

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