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Properties

Label 2-1053-9.4-c1-0-8
Degree $2$
Conductor $1053$
Sign $0.939 + 0.342i$
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  + (−0.809 − 1.40i)2-s + (−0.309 + 0.535i)4-s + (−1.30 + 2.26i)5-s + (−1.11 − 1.93i)7-s − 2.23·8-s + 4.23·10-s + (0.427 + 0.739i)11-s + (0.5 − 0.866i)13-s + (−1.80 + 3.13i)14-s + (2.42 + 4.20i)16-s + 4.61·17-s − 4.09·19-s + (−0.809 − 1.40i)20-s + (0.690 − 1.19i)22-s + (−3.42 + 5.93i)23-s + ⋯
L(s)  = 1  + (−0.572 − 0.990i)2-s + (−0.154 + 0.267i)4-s + (−0.585 + 1.01i)5-s + (−0.422 − 0.731i)7-s − 0.790·8-s + 1.33·10-s + (0.128 + 0.223i)11-s + (0.138 − 0.240i)13-s + (−0.483 + 0.837i)14-s + (0.606 + 1.05i)16-s + 1.12·17-s − 0.938·19-s + (−0.180 − 0.313i)20-s + (0.147 − 0.255i)22-s + (−0.714 + 1.23i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(8.40824\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8547812895\)
\(L(\frac12)\) \(\approx\) \(0.8547812895\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 1.40i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.11 + 1.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.427 - 0.739i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 + (3.42 - 5.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.35 - 4.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.23 + 7.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.23T + 37T^{2} \)
41 \( 1 + (-0.118 + 0.204i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.66 - 9.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.545 + 0.944i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.14T + 53T^{2} \)
59 \( 1 + (-3.11 + 5.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.59 - 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 - 4.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.291T + 71T^{2} \)
73 \( 1 - 3.38T + 73T^{2} \)
79 \( 1 + (1.66 + 2.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.97 - 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + (1.23 + 2.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.02570106590322773224680123082, −9.455981076906472004688212586582, −8.210871030846761009756302969797, −7.50304768491293138284832363190, −6.58004536521547902269746368751, −5.76167134723036797441883832825, −4.09232335375096601640881974097, −3.38574419793455908253200983173, −2.46930080795760690065653155111, −0.988458572960908512484822063893, 0.60228722596938259221548317735, 2.56508386823486719961682426452, 3.85427356478668893171780606440, 4.93622610225011186310293410642, 5.98373509508078561173658031372, 6.52857293439579162436400520968, 7.68222753494444547003909146391, 8.406599587980975541146937259817, 8.742731159673010426061096686965, 9.594921163410263239009185042473

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