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Properties

Label 2-189-63.31-c2-0-1
Degree $2$
Conductor $189$
Sign $-0.981 + 0.191i$
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  + (−0.826 + 1.43i)2-s + (0.633 + 1.09i)4-s + 7.86i·5-s + (−5.81 − 3.89i)7-s − 8.70·8-s + (−11.2 − 6.50i)10-s + 0.712·11-s + (11.3 + 6.54i)13-s + (10.3 − 5.11i)14-s + (4.66 − 8.08i)16-s + (−14.9 − 8.62i)17-s + (3.67 − 2.12i)19-s + (−8.63 + 4.98i)20-s + (−0.589 + 1.02i)22-s − 15.4·23-s + ⋯
L(s)  = 1  + (−0.413 + 0.715i)2-s + (0.158 + 0.274i)4-s + 1.57i·5-s + (−0.831 − 0.555i)7-s − 1.08·8-s + (−1.12 − 0.650i)10-s + 0.0647·11-s + (0.871 + 0.503i)13-s + (0.741 − 0.365i)14-s + (0.291 − 0.505i)16-s + (−0.878 − 0.507i)17-s + (0.193 − 0.111i)19-s + (−0.431 + 0.249i)20-s + (−0.0267 + 0.0463i)22-s − 0.672·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.981 + 0.191i$
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: \((2,\ 189,\ (\ :1),\ -0.981 + 0.191i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0786972 - 0.813522i\)
\(L(\frac12)\) \(\approx\) \(0.0786972 - 0.813522i\)
\(L(2)\) 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 \( 1 + (5.81 + 3.89i)T \)
good2 \( 1 + (0.826 - 1.43i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 - 7.86iT - 25T^{2} \)
11 \( 1 - 0.712T + 121T^{2} \)
13 \( 1 + (-11.3 - 6.54i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (14.9 + 8.62i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.67 + 2.12i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 15.4T + 529T^{2} \)
29 \( 1 + (-8.42 - 14.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (38.1 - 22.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-23.2 - 40.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-47.7 - 27.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (17.6 + 30.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.12 + 0.647i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (36.4 - 63.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-45.0 + 26.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-81.9 - 47.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-45.5 - 78.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 34.4T + 5.04e3T^{2} \)
73 \( 1 + (28.6 + 16.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (47.3 - 81.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-79.2 + 45.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-58.0 + 33.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-33.5 + 19.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.98758567376085536663785610518, −11.63243430852314378885389502522, −10.89138431088687664718475341235, −9.826537933960888010825734366657, −8.749153309784391670867779379961, −7.41022215987088642305885458122, −6.78793758581395180542736686157, −6.12224810457618215297528921057, −3.79770132008070562038319458181, −2.77415141965922749302211777455, 0.52338801115386586211604204484, 2.06725758685913131913537123719, 3.83618453998393801072898729140, 5.48153046813569489907233347369, 6.26110746986458518952226099708, 8.188192823714699475371962957698, 9.078683757263519496572156759664, 9.636023925769750217248716944537, 10.85210571757986091166166058197, 11.82325280064171171632339072670

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