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Properties

Label 2-1053-9.7-c1-0-41
Degree $2$
Conductor $1053$
Sign $-0.173 + 0.984i$
Analytic cond. $8.40824$
Root an. cond. $2.89969$
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  + (−1.18 + 2.05i)2-s + (−1.80 − 3.13i)4-s + (−1.91 − 3.32i)5-s + (2.23 − 3.87i)7-s + 3.83·8-s + 9.09·10-s + (1.46 − 2.53i)11-s + (0.5 + 0.866i)13-s + (5.30 + 9.17i)14-s + (−0.927 + 1.60i)16-s − 4.74·17-s + 0.763·19-s + (−6.93 + 12.0i)20-s + (3.47 + 6.01i)22-s + (−1.46 − 2.53i)23-s + ⋯
L(s)  = 1  + (−0.838 + 1.45i)2-s + (−0.904 − 1.56i)4-s + (−0.857 − 1.48i)5-s + (0.845 − 1.46i)7-s + 1.35·8-s + 2.87·10-s + (0.441 − 0.765i)11-s + (0.138 + 0.240i)13-s + (1.41 + 2.45i)14-s + (−0.231 + 0.401i)16-s − 1.14·17-s + 0.175·19-s + (−1.55 + 2.68i)20-s + (0.740 + 1.28i)22-s + (−0.305 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(8.40824\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5435578792\)
\(L(\frac12)\) \(\approx\) \(0.5435578792\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.18 - 2.05i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.91 + 3.32i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.23 + 3.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.46 + 2.53i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - 0.763T + 19T^{2} \)
23 \( 1 + (1.46 + 2.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.46 - 2.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.61 + 6.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 + (-2.37 - 4.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.118 - 0.204i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.91 + 3.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.85T + 53T^{2} \)
59 \( 1 + (-1.01 - 1.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.11 - 8.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 2.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.57T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.452 - 0.784i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (9.09 - 15.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.055166323601029030195062331440, −8.795148037531242195524405700449, −7.890170412385609916875521428097, −7.51383672791300870904131511187, −6.54152715567073209564623771993, −5.48429580084547733913459484269, −4.50064772938084154881627209683, −4.02682851137795906780012062918, −1.26477155379946770302571879664, −0.36330759833562923673251332691, 1.86985285697669205266118182323, 2.56082132068306201133851647740, 3.49787895586790825050861519400, 4.50055411093033704642778034782, 5.94700476194480065543413498315, 7.10703816456098397948190034142, 7.915369834303819117597041815596, 8.723664446024701433866380591987, 9.385997804350324787238030663224, 10.33703274732000107994519342283

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