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Properties

Label 2-1053-117.115-c0-0-0
Degree $2$
Conductor $1053$
Sign $0.790 + 0.612i$
Analytic cond. $0.525515$
Root an. cond. $0.724924$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·4-s + (1.36 + 0.366i)7-s + (0.866 − 0.5i)13-s − 16-s + (−1.86 + 0.5i)19-s + (0.866 + 0.5i)25-s + (0.366 − 1.36i)28-s + (0.133 − 0.5i)31-s + (−0.5 − 0.133i)37-s + (1.5 − 0.866i)43-s + (0.866 + 0.5i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s + i·64-s + (−1.86 + 0.5i)67-s + ⋯
L(s)  = 1  − i·4-s + (1.36 + 0.366i)7-s + (0.866 − 0.5i)13-s − 16-s + (−1.86 + 0.5i)19-s + (0.866 + 0.5i)25-s + (0.366 − 1.36i)28-s + (0.133 − 0.5i)31-s + (−0.5 − 0.133i)37-s + (1.5 − 0.866i)43-s + (0.866 + 0.5i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s + i·64-s + (−1.86 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(0.525515\)
Root analytic conductor: \(0.724924\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :0),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169896218\)
\(L(\frac12)\) \(\approx\) \(1.169896218\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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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

−10.26614352558357109390960450977, −9.004586211593924466257542584892, −8.556033315572340527200370477403, −7.62237877198390450413606418175, −6.44349799219986015869052521966, −5.72652759694820104339948829329, −4.91865824301696614113832841713, −4.02836833980096695020634244918, −2.34422098883461694228211026036, −1.36100195807167291870398411891, 1.69579698879086947381424681942, 2.92075635556168363370907075603, 4.33214069245162835833062651811, 4.52211205373209389539420001455, 6.07462461841598969185374974621, 6.99461958155489945749451669456, 7.78792149418219742388162403253, 8.654947338712099557387686845525, 8.888353585921958261506490808356, 10.52220573458455826012059203335

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