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Properties

Label 2-189-9.4-c1-0-4
Degree $2$
Conductor $189$
Sign $0.514 + 0.857i$
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  + (−0.119 − 0.207i)2-s + (0.971 − 1.68i)4-s + (0.590 − 1.02i)5-s + (0.5 + 0.866i)7-s − 0.942·8-s − 0.282·10-s + (−1.85 − 3.20i)11-s + (−0.5 + 0.866i)13-s + (0.119 − 0.207i)14-s + (−1.83 − 3.16i)16-s + 6.94·17-s + 1.94·19-s + (−1.14 − 1.98i)20-s + (−0.442 + 0.766i)22-s + (−2.80 + 4.85i)23-s + ⋯
L(s)  = 1  + (−0.0845 − 0.146i)2-s + (0.485 − 0.841i)4-s + (0.264 − 0.457i)5-s + (0.188 + 0.327i)7-s − 0.333·8-s − 0.0893·10-s + (−0.558 − 0.967i)11-s + (−0.138 + 0.240i)13-s + (0.0319 − 0.0553i)14-s + (−0.457 − 0.792i)16-s + 1.68·17-s + 0.445·19-s + (−0.256 − 0.444i)20-s + (−0.0944 + 0.163i)22-s + (−0.584 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10634 - 0.626457i\)
\(L(\frac12)\) \(\approx\) \(1.10634 - 0.626457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.119 + 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.590 + 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.85 + 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + (2.80 - 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (5.09 - 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-1.30 + 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 6.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 + (3.58 + 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.18481458758104423941083298143, −11.46056954701142528423273083574, −10.37347172104846352606767311281, −9.587018857775626102008780839334, −8.477692004343374430021661413847, −7.24857727456283483338327968032, −5.74539050755666355124291412163, −5.28305784153718558869712343548, −3.16314228315911781210588835565, −1.41710530139108223374200000506, 2.36787573357677827298945058940, 3.71026619993020913154436508201, 5.32023169211293971085883323480, 6.77926427235722744338350715869, 7.54226528091218168519797834400, 8.455214660501816704549666128549, 10.00172540720294089708261259804, 10.59866366337889658422000822089, 12.03956203602862854006344400817, 12.43500138911033343405462488753

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