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Properties

Label 2-1050-35.33-c1-0-6
Degree $2$
Conductor $1050$
Sign $-0.863 - 0.503i$
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 + (1.52 + 2.15i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (0.883 + 1.52i)11-s + (0.965 − 0.258i)12-s + (−2.71 + 2.71i)13-s + (−1.69 + 2.03i)14-s + (0.500 − 0.866i)16-s + (0.574 − 2.14i)17-s + (−0.258 + 0.965i)18-s + (0.886 − 1.53i)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 + (0.577 + 0.816i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.266 + 0.461i)11-s + (0.278 − 0.0747i)12-s + (−0.752 + 0.752i)13-s + (−0.451 + 0.543i)14-s + (0.125 − 0.216i)16-s + (0.139 − 0.520i)17-s + (−0.0610 + 0.227i)18-s + (0.203 − 0.352i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.863 - 0.503i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.103273610\)
\(L(\frac12)\) \(\approx\) \(1.103273610\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 2.15i)T \)
good11 \( 1 + (-0.883 - 1.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.71 - 2.71i)T - 13iT^{2} \)
17 \( 1 + (-0.574 + 2.14i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.886 + 1.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.90 + 1.04i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.84iT - 29T^{2} \)
31 \( 1 + (8.94 - 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.861 - 3.21i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (3.46 + 3.46i)T + 43iT^{2} \)
47 \( 1 + (5.93 - 1.59i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.106 - 0.396i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.18 - 8.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.87 + 3.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.37 - 1.97i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (10.2 + 2.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.9 + 6.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 + 1.94i)T - 83iT^{2} \)
89 \( 1 + (0.558 - 0.966i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \)
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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

−10.13859380590569996733979601180, −9.228773565103525094482303907866, −8.658351001308217235868220402834, −7.45351937692670388672609445666, −6.97638087818268436654669876658, −6.00878988674747290815000300112, −4.99128330849254717365739842729, −4.66371556492397404160358067015, −3.05347235481771540499041450922, −1.63533537418625158125020099659, 0.51443863736795240421692658149, 1.85085545169642869027110107295, 3.36595741452718448294634323111, 4.17767958014350161657877029347, 5.18042082517450261803387907399, 5.86094398487392349287508437238, 7.13235756631481505762844848458, 7.86242609243097203358770952975, 8.911311722286820824186183721848, 9.892746649135736190629811553883

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