Properties

Degree 2
Conductor $ 3^{3} \cdot 7 $
Sign $0.574 + 0.818i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.730 − 1.26i)2-s + (−0.0665 − 0.115i)4-s + (0.296 − 0.514i)5-s + (2.32 + 1.26i)7-s + 2.72·8-s + (−0.433 − 0.750i)10-s + (−2.23 − 3.86i)11-s − 4.51·13-s + (3.29 − 2.01i)14-s + (2.12 − 3.67i)16-s + (0.136 + 0.236i)17-s + (−1.43 + 2.48i)19-s − 0.0789·20-s − 6.51·22-s + (−2.52 + 4.37i)23-s + ⋯
L(s)  = 1  + (0.516 − 0.894i)2-s + (−0.0332 − 0.0576i)4-s + (0.132 − 0.229i)5-s + (0.878 + 0.478i)7-s + 0.964·8-s + (−0.137 − 0.237i)10-s + (−0.672 − 1.16i)11-s − 1.25·13-s + (0.881 − 0.538i)14-s + (0.531 − 0.919i)16-s + (0.0331 + 0.0574i)17-s + (−0.328 + 0.569i)19-s − 0.0176·20-s − 1.38·22-s + (−0.526 + 0.912i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(189\)    =    \(3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.574 + 0.818i$
motivic weight  =  \(1\)
character  :  $\chi_{189} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 189,\ (\ :1/2),\ 0.574 + 0.818i)$
$L(1)$  $\approx$  $1.48105 - 0.769802i$
$L(\frac12)$  $\approx$  $1.48105 - 0.769802i$
$L(\frac{3}{2})$   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 \( 1 + (-2.32 - 1.26i)T \)
good2 \( 1 + (-0.730 + 1.26i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.296 + 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.23 + 3.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
17 \( 1 + (-0.136 - 0.236i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.43 - 2.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.52 - 4.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.352T + 29T^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.32 + 5.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 3.38T + 43T^{2} \)
47 \( 1 + (6.21 - 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.66 + 9.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.02 - 6.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.36 + 2.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.93 + 5.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.60T + 71T^{2} \)
73 \( 1 + (5.55 + 9.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.58 - 9.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + (-2.68 + 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−12.35286126164403648332282772689, −11.49420622459907807941323074091, −10.82088478441631392524790274220, −9.666873093922442376984319968456, −8.302962994162382241992868589100, −7.51102597759450757616874456943, −5.65753152070042961390964534084, −4.72735538671741628648836767723, −3.24238314136025927764919275467, −1.92959202178053118598540503962, 2.18925864776931347484174428161, 4.56370563194415387173852405404, 5.04616122491773614662085498031, 6.62151194074563675045925169339, 7.34614281512022805531858965714, 8.271887336887227106650469457814, 10.03177039228435817918511922381, 10.53595193207495602594035410119, 11.87092423899478930050305658426, 12.93969963374284468405716770875

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