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Properties

Label 2-1053-117.77-c0-0-4
Degree $2$
Conductor $1053$
Sign $0.984 - 0.173i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)5-s + 8-s + 0.999·10-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)16-s + (0.999 − 1.73i)22-s − 0.999·26-s + (0.500 − 0.866i)40-s + (−1 + 1.73i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s − 1.99·55-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)5-s + 8-s + 0.999·10-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)16-s + (0.999 − 1.73i)22-s − 0.999·26-s + (0.500 − 0.866i)40-s + (−1 + 1.73i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s − 1.99·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.525515\)
Root analytic conductor: \(0.724924\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490409801\)
\(L(\frac12)\) \(\approx\) \(1.490409801\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.06942820680872780872066470892, −9.171565978688312210715547109887, −8.332487965172690128004688832314, −7.62403447600984565853033759032, −6.51441254598128860078775004425, −5.82547067056910800954462127849, −5.14392376538708454835420093869, −4.38448023242321106480964278464, −2.89947693122478674799578662363, −1.39496115092482000545076536337, 2.08247632740122179088949175540, 2.57001650902277579861939033167, 3.66279351297684850678803550861, 4.78472481913722713908498272937, 5.53020961113711239816994029076, 7.03829952626775515254620551683, 7.29869128843745592894245542809, 8.370343583012372506211280103272, 9.798582805586021469146768845968, 10.31809239928207039048761159160

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