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Properties

Label 2-1050-35.4-c1-0-17
Degree $2$
Conductor $1050$
Sign $-0.441 + 0.897i$
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  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−0.866 + 2.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1 − 1.73i)11-s + (−0.866 + 0.499i)12-s − 4i·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−4.33 − 2.5i)17-s + (0.866 + 0.499i)18-s + (−2 − 3.46i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (−0.327 + 0.944i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s + (−0.249 + 0.144i)12-s − 1.10i·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−1.05 − 0.606i)17-s + (0.204 + 0.117i)18-s + (−0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.541336464\)
\(L(\frac12)\) \(\approx\) \(1.541336464\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 - 2.5i)T \)
good11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.33 + 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (1.73 + i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−9.736639216217508647996985829354, −8.879066153605457782529677333291, −8.028063827771741077000361669105, −6.68515767428171240815137792316, −6.27078368481508208528476928772, −5.25761743931894253841530279681, −4.56032970873753102977634176512, −3.10093320150734230222297993102, −2.36232564205064519064692102685, −0.60809531005130475931922103379, 1.59948932315091214143515757561, 3.26126471658803208745443960916, 4.32871316566124276510748097831, 4.68745019040997795796992304092, 6.08292712785656322092625826152, 6.71807671915113362617175386803, 7.29853002975653046513195184013, 8.546229775557506339830088937443, 9.349216646049414639850759894051, 10.42802241030279830643472581846

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