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Properties

Label 2-57-19.11-c1-0-1
Degree $2$
Conductor $57$
Sign $0.813 - 0.582i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + (0.499 + 0.866i)6-s + 7-s − 3·8-s + (−0.499 − 0.866i)9-s − 2·11-s + 12-s + (−2.5 − 4.33i)13-s + (−0.5 + 0.866i)14-s + (0.500 − 0.866i)16-s + (2 − 3.46i)17-s + 0.999·18-s + (−4 + 1.73i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s + (0.204 + 0.353i)6-s + 0.377·7-s − 1.06·8-s + (−0.166 − 0.288i)9-s − 0.603·11-s + 0.288·12-s + (−0.693 − 1.20i)13-s + (−0.133 + 0.231i)14-s + (0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + 0.235·18-s + (−0.917 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.813 - 0.582i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775770 + 0.249138i\)
\(L(\frac12)\) \(\approx\) \(0.775770 + 0.249138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.35860856735610309898704050614, −14.50373110497224869643878491657, −12.98621670853570510070654685970, −12.20507992973222323767465292571, −10.75341689053599515996358739339, −9.097133540483271614089070009842, −7.894860116879307797435495500975, −7.18240199484817905418660626402, −5.48594606438498548166885130784, −2.94946291741863806250225006895, 2.40818147527031805418881116977, 4.60961511505194332926976804502, 6.38696664656561744722230314604, 8.262668132440760944071462396690, 9.513488209222794924894177986716, 10.48221575496679757289342798633, 11.42171316795461322732629094461, 12.67371856647351296888438461030, 14.34373906974706481615562195229, 14.95904509442818236902213106077

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