Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.799 - 0.601i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.886 + 0.322i)2-s + (−0.292 + 1.70i)3-s + (−0.850 − 0.713i)4-s + (0.485 + 0.578i)5-s + (−0.809 + 1.41i)6-s + (1.38 − 2.39i)7-s + (−1.46 − 2.54i)8-s + (−2.82 − 0.997i)9-s + (0.243 + 0.669i)10-s + (−2.28 + 1.31i)11-s + (1.46 − 1.24i)12-s + (1.90 + 0.335i)13-s + (1.99 − 1.67i)14-s + (−1.13 + 0.660i)15-s + (−0.0947 − 0.537i)16-s + (0.220 − 0.606i)17-s + ⋯
L(s)  = 1  + (0.626 + 0.228i)2-s + (−0.168 + 0.985i)3-s + (−0.425 − 0.356i)4-s + (0.217 + 0.258i)5-s + (−0.330 + 0.579i)6-s + (0.522 − 0.905i)7-s + (−0.518 − 0.898i)8-s + (−0.943 − 0.332i)9-s + (0.0770 + 0.211i)10-s + (−0.687 + 0.396i)11-s + (0.423 − 0.359i)12-s + (0.527 + 0.0930i)13-s + (0.534 − 0.448i)14-s + (−0.291 + 0.170i)15-s + (−0.0236 − 0.134i)16-s + (0.0535 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.799 - 0.601i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.799 - 0.601i)$
$L(1)$  $\approx$  $0.950776 + 0.317689i$
$L(\frac12)$  $\approx$  $0.950776 + 0.317689i$
$L(\frac{3}{2})$   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 \( 1 + (0.292 - 1.70i)T \)
19 \( 1 + (1.94 - 3.90i)T \)
good2 \( 1 + (-0.886 - 0.322i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.485 - 0.578i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.28 - 1.31i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.90 - 0.335i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.220 + 0.606i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (5.61 - 6.69i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.33 + 3.03i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.09iT - 37T^{2} \)
41 \( 1 + (0.930 + 5.27i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.12 - 0.947i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (3.48 + 9.56i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.16 + 3.49i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-7.06 - 2.57i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.37 + 1.15i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.35 + 9.20i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.08 - 1.74i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.952 - 5.39i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-9.22 + 1.62i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-7.02 - 4.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.233 + 1.32i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.64 + 12.7i)T + (-74.3 - 62.3i)T^{2} \)
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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

−15.27469970438643501195450773297, −14.17887535653483333184676965315, −13.62578466929609396123676511143, −11.95787477800527420094023711862, −10.42797705697286736403324964968, −9.940624361556344808872312343225, −8.238999125872871437091250287437, −6.28107073175052526795200706681, −4.96062605106334285854110791589, −3.84256181240069650223152206107, 2.61813033520753190504151594410, 4.93999522531040854838516867441, 6.12635160791796877658300468145, 8.107888411161802633793631612891, 8.784712059537361134272356116189, 10.99420061912256143056176155470, 12.10109376327517736420541434099, 12.86051265833739446537669908417, 13.73235697552766500059522601260, 14.73362286852160524584237218640

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