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Properties

Label 2-1053-117.31-c0-0-0
Degree $2$
Conductor $1053$
Sign $0.546 - 0.837i$
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  + (−0.866 − 0.5i)4-s + (−0.366 + 1.36i)7-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (−1 + i)19-s + (0.866 − 0.5i)25-s + (1 − 0.999i)28-s + (0.366 + 1.36i)31-s + (1 + i)37-s + (−0.866 − 0.5i)49-s + (−0.499 − 0.866i)52-s − 0.999i·64-s + (−0.366 − 1.36i)67-s + (−1 − i)73-s + (1.36 − 0.366i)76-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)4-s + (−0.366 + 1.36i)7-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (−1 + i)19-s + (0.866 − 0.5i)25-s + (1 − 0.999i)28-s + (0.366 + 1.36i)31-s + (1 + i)37-s + (−0.866 − 0.5i)49-s + (−0.499 − 0.866i)52-s − 0.999i·64-s + (−0.366 − 1.36i)67-s + (−1 − i)73-s + (1.36 − 0.366i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(0.525515\)
Root analytic conductor: \(0.724924\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :0),\ 0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7691394746\)
\(L(\frac12)\) \(\approx\) \(0.7691394746\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−10.16960218165526213174042703201, −9.267443723357738505465232278907, −8.693507075294842602197756346098, −8.149639942621242446875973590636, −6.50016986121600570142966358012, −6.05684049972813905510002596884, −5.09483373265622154232903231747, −4.18745830780474049514533370521, −3.00308334381189184732855588110, −1.60720037013402787442658105614, 0.794864837539263430007271472500, 2.85903625350512672689144687492, 3.94853676393762341947859947883, 4.42567213703835788186160060856, 5.66557112231000737105543865605, 6.76262068206111734104709958169, 7.53266559401714879809711529465, 8.356340379979551369493475703969, 9.122610707583587720186651844354, 9.949247506476269487686435030098

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