Properties

Label 2-189-7.2-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.605 + 0.795i$
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  + (−1.22 + 2.12i)2-s + (−1.99 − 3.46i)4-s + (−1.22 + 2.12i)5-s + (−0.5 + 2.59i)7-s + 4.89·8-s + (−2.99 − 5.19i)10-s + (−2.44 − 4.24i)11-s − 4·13-s + (−4.89 − 4.24i)14-s + (−1.99 + 3.46i)16-s + (1.22 + 2.12i)17-s + (0.5 − 0.866i)19-s + 9.79·20-s + 11.9·22-s + (−1.22 + 2.12i)23-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)2-s + (−0.999 − 1.73i)4-s + (−0.547 + 0.948i)5-s + (−0.188 + 0.981i)7-s + 1.73·8-s + (−0.948 − 1.64i)10-s + (−0.738 − 1.27i)11-s − 1.10·13-s + (−1.30 − 1.13i)14-s + (−0.499 + 0.866i)16-s + (0.297 + 0.514i)17-s + (0.114 − 0.198i)19-s + 2.19·20-s + 2.55·22-s + (−0.255 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153342 - 0.309329i\)
\(L(\frac12)\) \(\approx\) \(0.153342 - 0.309329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + (1.22 - 2.12i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.34T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (1.22 - 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.22 - 2.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.89 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (1.22 - 2.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−13.57399177719301512157896685485, −12.13294963529034038110259133451, −10.97022646758096752872811490228, −9.961721899768226859768992188906, −8.898919307860631113655323304460, −8.007457901723296254136879527838, −7.20533030033003340705467831477, −6.10988063794248592995902091945, −5.25733256871539093432632858317, −3.04210186654321559409953884970, 0.39402985270323709124190149943, 2.22569423766529871921410543048, 3.90766409721401139967098081221, 4.87215554695444942860836066911, 7.38437923010358338865086347610, 7.994620343981560846480658472270, 9.386107836959756905613687760093, 9.913298887405308985964321552395, 10.85014101074360108839299652341, 11.96398463653769631447124868982

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