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Properties

Label 2-189-63.31-c2-0-10
Degree $2$
Conductor $189$
Sign $-0.110 + 0.993i$
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  + (1.12 − 1.94i)2-s + (−0.509 − 0.883i)4-s − 1.93i·5-s + (3.87 − 5.83i)7-s + 6.67·8-s + (−3.75 − 2.16i)10-s + 0.372·11-s + (−5.01 − 2.89i)13-s + (−6.98 − 14.0i)14-s + (9.51 − 16.4i)16-s + (−9.96 − 5.75i)17-s + (18.2 − 10.5i)19-s + (−1.70 + 0.986i)20-s + (0.417 − 0.722i)22-s − 27.2·23-s + ⋯
L(s)  = 1  + (0.560 − 0.970i)2-s + (−0.127 − 0.220i)4-s − 0.386i·5-s + (0.552 − 0.833i)7-s + 0.834·8-s + (−0.375 − 0.216i)10-s + 0.0338·11-s + (−0.386 − 0.222i)13-s + (−0.498 − 1.00i)14-s + (0.594 − 1.03i)16-s + (−0.586 − 0.338i)17-s + (0.959 − 0.554i)19-s + (−0.0854 + 0.0493i)20-s + (0.0189 − 0.0328i)22-s − 1.18·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.110 + 0.993i$
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: \((2,\ 189,\ (\ :1),\ -0.110 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47624 - 1.64905i\)
\(L(\frac12)\) \(\approx\) \(1.47624 - 1.64905i\)
\(L(2)\) 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 \( 1 + (-3.87 + 5.83i)T \)
good2 \( 1 + (-1.12 + 1.94i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + 1.93iT - 25T^{2} \)
11 \( 1 - 0.372T + 121T^{2} \)
13 \( 1 + (5.01 + 2.89i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (9.96 + 5.75i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-18.2 + 10.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 27.2T + 529T^{2} \)
29 \( 1 + (-20.1 - 34.8i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (42.2 - 24.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-14.7 - 25.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-19.6 - 11.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-10.7 - 18.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-46.1 - 26.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (43.7 - 75.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-7.99 + 4.61i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (61.0 + 35.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-37.7 - 65.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 97.4T + 5.04e3T^{2} \)
73 \( 1 + (75.3 + 43.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-23.5 + 40.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (81.1 - 46.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-25.6 + 14.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (122. - 70.7i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.13568731718314480392790432121, −11.13214047632367113647618351440, −10.50765281809927601127901769372, −9.288696879277852292039912095202, −7.920598875102376907615633702987, −7.01530239049466850126537620147, −5.12277431047529445489803955453, −4.29148132826939929276260028880, −2.93385699486150318121883358015, −1.29195653443941160658645526466, 2.13660842258651718428915940955, 4.12650335878123857355705047764, 5.37599578493423420391583087465, 6.17490251943507703078120707420, 7.33786603945067645652331078184, 8.221758371048524182989323505808, 9.537683125402642210216134744983, 10.72342428127968933499387891544, 11.70053517650296692071697567453, 12.73646638465802905788798470574

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