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Properties

Label 2-1050-35.9-c1-0-21
Degree $2$
Conductor $1050$
Sign $-0.753 + 0.657i$
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 + (−1.5 − 2.59i)11-s + (0.866 + 0.499i)12-s − 4i·13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (−2 + 3.46i)19-s + (−0.500 − 2.59i)21-s + 3i·22-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 + (−0.452 − 0.783i)11-s + (0.249 + 0.144i)12-s − 1.10i·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.204 + 0.117i)18-s + (−0.458 + 0.794i)19-s + (−0.109 − 0.566i)21-s + 0.639i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.119959802\)
\(L(\frac12)\) \(\approx\) \(1.119959802\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.681010565272835361810028282006, −8.623525325609736620343020917750, −7.963102286541721398347029468090, −7.48973332233434193310871220290, −6.38079315795562488332181169750, −5.30125367167776825271217438093, −3.91146858391424674319673687106, −3.16931762337912344316148616245, −1.88143411264106142644343491001, −0.56198224361976666049644586917, 1.83520649036739363113886438926, 2.63371628565138776713312334180, 4.21269898922948256031232087655, 5.06123149301111033998771767411, 6.08485867848573934995128694992, 7.07028578368954141668226250585, 7.85848916848981913840704747734, 8.668972833192047773906647936589, 9.400249910035392433448079905157, 9.798608787261531390001420104923

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