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Properties

Label 2-189-63.25-c1-0-5
Degree $2$
Conductor $189$
Sign $0.983 - 0.182i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + 2.38·2-s + 3.69·4-s + (−1.46 + 2.52i)5-s + (−0.138 − 2.64i)7-s + 4.05·8-s + (−3.48 + 6.03i)10-s + (−0.676 − 1.17i)11-s + (−0.733 − 1.26i)13-s + (−0.330 − 6.30i)14-s + 2.27·16-s + (−1.65 + 2.86i)17-s + (−1.10 − 1.91i)19-s + (−5.39 + 9.35i)20-s + (−1.61 − 2.79i)22-s + (1.31 − 2.27i)23-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.84·4-s + (−0.653 + 1.13i)5-s + (−0.0523 − 0.998i)7-s + 1.43·8-s + (−1.10 + 1.90i)10-s + (−0.204 − 0.353i)11-s + (−0.203 − 0.352i)13-s + (−0.0883 − 1.68i)14-s + 0.568·16-s + (−0.401 + 0.695i)17-s + (−0.253 − 0.438i)19-s + (−1.20 + 2.09i)20-s + (−0.344 − 0.596i)22-s + (0.274 − 0.474i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.983 - 0.182i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46367 + 0.226450i\)
\(L(\frac12)\) \(\approx\) \(2.46367 + 0.226450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.138 + 2.64i)T \)
good2 \( 1 - 2.38T + 2T^{2} \)
5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.676 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.733 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.10 + 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.521 - 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.904 - 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.97T + 47T^{2} \)
53 \( 1 + (-3.22 + 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 0.559T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.767T + 79T^{2} \)
83 \( 1 + (-0.983 + 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.20 + 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.14 - 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.92860517109563270801519812148, −11.62366028669013314807228622738, −11.02828419611295588321204321418, −10.18931373718056202572058454121, −8.108040667549840257939670287581, −6.97100462461788598646928278002, −6.34392998516726295123224008965, −4.79391543438098842522958777063, −3.75844561925907912323723201425, −2.81593262795084848068713332612, 2.40059731473736897453004551597, 3.98772343880079845259406960332, 4.92054284182461805655849682452, 5.73897066403715048473192852532, 7.08798778073465736468950056317, 8.414258267649646609343281786472, 9.481543599625953815089241097231, 11.24531742094115863617584778001, 11.98129430808715641295444244805, 12.56669067070254788198295707452

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