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Properties

Label 2-1050-5.2-c2-0-7
Degree $2$
Conductor $1050$
Sign $-0.973 + 0.229i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s − 2.44·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s − 5.03·11-s + (−2.44 − 2.44i)12-s + (2.44 − 2.44i)13-s + 3.74i·14-s − 4·16-s + (18.2 + 18.2i)17-s + (2.99 − 2.99i)18-s + 9.56i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s − 0.408·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s − 0.457·11-s + (−0.204 − 0.204i)12-s + (0.188 − 0.188i)13-s + 0.267i·14-s − 0.250·16-s + (1.07 + 1.07i)17-s + (0.166 − 0.166i)18-s + 0.503i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.187861353\)
\(L(\frac12)\) \(\approx\) \(1.187861353\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 + 5.03T + 121T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 169iT^{2} \)
17 \( 1 + (-18.2 - 18.2i)T + 289iT^{2} \)
19 \( 1 - 9.56iT - 361T^{2} \)
23 \( 1 + (16.4 - 16.4i)T - 529iT^{2} \)
29 \( 1 + 4.18iT - 841T^{2} \)
31 \( 1 + 55.1T + 961T^{2} \)
37 \( 1 + (-1.23 - 1.23i)T + 1.36e3iT^{2} \)
41 \( 1 + 12.2T + 1.68e3T^{2} \)
43 \( 1 + (36.1 - 36.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (18.6 + 18.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (37.8 - 37.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 60.8T + 3.72e3T^{2} \)
67 \( 1 + (30.4 + 30.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 115.T + 5.04e3T^{2} \)
73 \( 1 + (54.8 - 54.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (52.8 - 52.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 16.5iT - 7.92e3T^{2} \)
97 \( 1 + (71.1 + 71.1i)T + 9.40e3iT^{2} \)
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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

−10.17711803152263548676535470780, −9.387699401787774205258755013960, −8.227554933175076637085101923639, −7.78233697251974404222789976581, −6.59654769520395040513520149056, −5.67532266931411463909902886418, −5.26376911632850751427880743904, −4.03296837385873663187170991535, −3.29271585831690959736178502557, −1.70271263967646018753449377272, 0.31703376874350775776968685707, 1.62039288656075999900312458039, 2.78380168444246770631734251325, 3.90811894851532615409757961151, 5.02456624443778069016340692788, 5.59157969616808513861475499978, 6.72874155202392213118953264097, 7.45035109707551900919330049654, 8.405811876917347372289114889257, 9.488517096517980020999245108756

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