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Properties

Label 2-1053-9.4-c1-0-12
Degree $2$
Conductor $1053$
Sign $-0.766 + 0.642i$
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  + (1.20 + 2.09i)2-s + (−1.91 + 3.31i)4-s + (−1.41 + 2.44i)5-s + (1.41 + 2.44i)7-s − 4.41·8-s − 6.82·10-s + (1 + 1.73i)11-s + (0.5 − 0.866i)13-s + (−3.41 + 5.91i)14-s + (−1.49 − 2.59i)16-s − 3.65·17-s + 2.82·19-s + (−5.41 − 9.37i)20-s + (−2.41 + 4.18i)22-s + (2 − 3.46i)23-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.957 + 1.65i)4-s + (−0.632 + 1.09i)5-s + (0.534 + 0.925i)7-s − 1.56·8-s − 2.15·10-s + (0.301 + 0.522i)11-s + (0.138 − 0.240i)13-s + (−0.912 + 1.58i)14-s + (−0.374 − 0.649i)16-s − 0.886·17-s + 0.648·19-s + (−1.21 − 2.09i)20-s + (−0.514 + 0.891i)22-s + (0.417 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 1053,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.120585575\)
\(L(\frac12)\) \(\approx\) \(2.120585575\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.41 - 2.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.41 + 5.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + (5.41 - 9.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.82 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.171 - 0.297i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-1.82 + 3.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.65 - 8.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.585 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + (5.65 + 9.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.82 - 6.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.17T + 89T^{2} \)
97 \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46835972961785291747595187687, −9.268621873159362624884828466082, −8.322821904495593475090486032213, −7.74984562626546365706673013643, −6.85062812474068281997134737709, −6.37062448759988580936562266392, −5.33834253117311400815916410242, −4.53728428351395401629707022482, −3.58078653369610791702588306013, −2.44534883060535764075915599123, 0.77244955387151828656093101702, 1.63889493769614887654267736238, 3.20912441611431608167577365191, 4.03277231630527918776244654718, 4.69974980131245745962409037361, 5.36948367860488835685257758147, 6.80348239683497819347426909173, 7.936401325871520443129849173635, 8.800678384327272359883352881690, 9.584952066479522763368681936085

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