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Properties

Label 2-189-63.31-c2-0-9
Degree $2$
Conductor $189$
Sign $0.963 - 0.268i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.198 + 0.343i)2-s + (1.92 + 3.32i)4-s − 2.97i·5-s + (2.98 − 6.33i)7-s − 3.10·8-s + (1.01 + 0.588i)10-s + 18.6·11-s + (9.50 + 5.48i)13-s + (1.58 + 2.27i)14-s + (−7.07 + 12.2i)16-s + (−5.75 − 3.32i)17-s + (−19.6 + 11.3i)19-s + (9.89 − 5.71i)20-s + (−3.69 + 6.40i)22-s + 26.9·23-s + ⋯
L(s)  = 1  + (−0.0990 + 0.171i)2-s + (0.480 + 0.832i)4-s − 0.594i·5-s + (0.426 − 0.904i)7-s − 0.388·8-s + (0.101 + 0.0588i)10-s + 1.69·11-s + (0.731 + 0.422i)13-s + (0.112 + 0.162i)14-s + (−0.441 + 0.765i)16-s + (−0.338 − 0.195i)17-s + (−1.03 + 0.597i)19-s + (0.494 − 0.285i)20-s + (−0.168 + 0.291i)22-s + 1.17·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.963 - 0.268i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73813 + 0.237536i\)
\(L(\frac12)\) \(\approx\) \(1.73813 + 0.237536i\)
\(L(2)\) 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 \( 1 + (-2.98 + 6.33i)T \)
good2 \( 1 + (0.198 - 0.343i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + 2.97iT - 25T^{2} \)
11 \( 1 - 18.6T + 121T^{2} \)
13 \( 1 + (-9.50 - 5.48i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (5.75 + 3.32i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (19.6 - 11.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 26.9T + 529T^{2} \)
29 \( 1 + (-10.3 - 17.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-16.4 + 9.49i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (3.57 + 6.19i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (66.6 + 38.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (14.9 + 25.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.23 + 3.02i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (19.9 - 34.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (30.4 - 17.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (27.8 + 16.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.778 + 1.34i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 111.T + 5.04e3T^{2} \)
73 \( 1 + (-37.1 - 21.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (22.9 - 39.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (31.3 - 18.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-35.9 + 20.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-120. + 69.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.29175283999678739194444848731, −11.49341027518499275128468461259, −10.60359605535939980899387227719, −8.979511061070657189766693332523, −8.500631165991845258285493460383, −7.11225479331561792928900627103, −6.44207525771521128935072916536, −4.53192683767387527208906829637, −3.59899978671289112569538281145, −1.46480661795539656589425260348, 1.48635417363047245123565799073, 2.97725364782034640431704805022, 4.78832522522894646142011460883, 6.26487444965864091854060816600, 6.68906042059864089916324209839, 8.511685085531015689004577232690, 9.280079099754148679071906080264, 10.51647267563163897133315582107, 11.29857263267698155645179771508, 11.93299068527919301991095413211

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