Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.813 + 0.582i$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.813 + 0.582i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.813 + 0.582i)$
$L(1)$  $\approx$  $0.775770 - 0.249138i$
$L(\frac12)$  $\approx$  $0.775770 - 0.249138i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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