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Properties

Label 2-189-189.101-c1-0-19
Degree $2$
Conductor $189$
Sign $-0.0499 + 0.998i$
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  + (1.47 − 1.76i)2-s + (1.66 − 0.480i)3-s + (−0.573 − 3.24i)4-s + (−1.41 + 1.18i)5-s + (1.61 − 3.64i)6-s + (−2.40 + 1.11i)7-s + (−2.59 − 1.49i)8-s + (2.53 − 1.59i)9-s + 4.24i·10-s + (−0.0789 + 0.0940i)11-s + (−2.51 − 5.13i)12-s + (0.0159 − 0.0437i)13-s + (−1.58 + 5.87i)14-s + (−1.78 + 2.65i)15-s + (−0.271 + 0.0988i)16-s + 0.157·17-s + ⋯
L(s)  = 1  + (1.04 − 1.24i)2-s + (0.960 − 0.277i)3-s + (−0.286 − 1.62i)4-s + (−0.631 + 0.529i)5-s + (0.659 − 1.48i)6-s + (−0.907 + 0.420i)7-s + (−0.916 − 0.528i)8-s + (0.845 − 0.533i)9-s + 1.34i·10-s + (−0.0238 + 0.0283i)11-s + (−0.726 − 1.48i)12-s + (0.00441 − 0.0121i)13-s + (−0.424 + 1.57i)14-s + (−0.459 + 0.684i)15-s + (−0.0678 + 0.0247i)16-s + 0.0381·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0499 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0499 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.0499 + 0.998i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.0499 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50321 - 1.58020i\)
\(L(\frac12)\) \(\approx\) \(1.50321 - 1.58020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.480i)T \)
7 \( 1 + (2.40 - 1.11i)T \)
good2 \( 1 + (-1.47 + 1.76i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.41 - 1.18i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.0789 - 0.0940i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.0159 + 0.0437i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 0.157T + 17T^{2} \)
19 \( 1 - 7.59iT - 19T^{2} \)
23 \( 1 + (-0.332 + 0.913i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.33 + 9.16i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.49 - 0.792i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.68 - 8.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.76 + 2.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.67 + 9.52i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.723 + 4.10i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.141 - 0.0818i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.1 - 4.04i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.98 - 0.527i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.53 - 6.32i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.863 - 0.498i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.95 + 4.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.99 - 1.67i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.02 + 2.19i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (13.2 + 2.33i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.30089029478815877551689123973, −11.76119424406042086555762680465, −10.39297603907269935213965239434, −9.727273787726688881075282767569, −8.351630577116544848438874883036, −7.10990506996625338216900620175, −5.71665495529589088212278133911, −3.90386104439652265780042285807, −3.35781542022104947419826673441, −2.09101461644161504646533878806, 3.22655976314610917410075192893, 4.17817259706919637171728487238, 5.18412077419297638536835869267, 6.79123917206615123883042751594, 7.43859412323156216807243281816, 8.553833309064195558850576701401, 9.447572586907925436503114765050, 10.89433563401290808392386720304, 12.57294006651403482619957370482, 13.03301902062797179044188528483

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