Properties

Degree 2
Conductor $ 3^{3} \cdot 7 $
Sign $-0.757 + 0.653i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.380 − 0.658i)2-s + (0.710 − 1.23i)4-s + (−1.59 − 2.75i)5-s + (−2.56 + 0.658i)7-s − 2.60·8-s + (−1.21 + 2.09i)10-s + (−1.11 + 1.93i)11-s + 3.70·13-s + (1.40 + 1.43i)14-s + (−0.430 − 0.746i)16-s + (2.80 − 4.85i)17-s + (−2.21 − 3.82i)19-s − 4.52·20-s + 1.70·22-s + (0.471 + 0.816i)23-s + ⋯
L(s)  = 1  + (−0.269 − 0.465i)2-s + (0.355 − 0.615i)4-s + (−0.711 − 1.23i)5-s + (−0.968 + 0.249i)7-s − 0.920·8-s + (−0.382 + 0.663i)10-s + (−0.337 + 0.584i)11-s + 1.02·13-s + (0.376 + 0.384i)14-s + (−0.107 − 0.186i)16-s + (0.679 − 1.17i)17-s + (−0.507 − 0.878i)19-s − 1.01·20-s + 0.363·22-s + (0.0982 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(189\)    =    \(3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.757 + 0.653i$
motivic weight  =  \(1\)
character  :  $\chi_{189} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 189,\ (\ :1/2),\ -0.757 + 0.653i)$
$L(1)$  $\approx$  $0.278983 - 0.750354i$
$L(\frac12)$  $\approx$  $0.278983 - 0.750354i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 + (2.56 - 0.658i)T \)
good2 \( 1 + (0.380 + 0.658i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.11 - 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.70T + 13T^{2} \)
17 \( 1 + (-2.80 + 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 3.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.471 - 0.816i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + (-2.85 + 4.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.56 + 2.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.98T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 + (-0.112 - 0.195i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.33 - 9.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.02 + 1.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.92 - 5.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.71 - 6.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + (3.77 - 6.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.41 - 5.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.11T + 83T^{2} \)
89 \( 1 + (4.86 + 8.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.842T + 97T^{2} \)
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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

−12.13788996373488374467018924002, −11.32255209022962493210968773268, −10.11461738277046303281159931177, −9.286841450195114863319041773209, −8.436262809477058063217872831626, −6.97789496896642725653303542541, −5.75937009812810085312094954301, −4.52901464755819055874357286326, −2.83725192711501270840058828502, −0.789988307387670108050688681938, 3.08503916168646420411761682599, 3.69153561356004538361060102442, 6.22894900122790426762571758586, 6.62750669789568535006674279058, 7.903453258495766537455818247838, 8.526093124279887150220772190738, 10.22029121603948213788998953151, 10.85572345543298044306077048162, 11.98032509411326382501995732627, 12.81420294707857093075471484186

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