Properties

Label 2-189-7.4-c1-0-3
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: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789900 + 1.59341i\)
\(L(\frac12)\) \(\approx\) \(0.789900 + 1.59341i\)
\(L(\frac{3}{2})\) 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

−13.54067783652982157923554419864, −12.28572656341934930708937920279, −10.94647435286020024635283465488, −9.901328454573310241433489773297, −8.496341649764835168194205485844, −7.35260581050469289555805905078, −6.60590713056930433224715436838, −5.84067678402851654791515757030, −4.40056290502220437786994293229, −3.21494881838272153135875681200, 1.67767735000749869672040150176, 2.85818645579476803603353314665, 4.73718544340895095236392094312, 5.04974381995710404371216687153, 6.68920028234694589583250605058, 8.741912310300545181441345269219, 9.586716397881844047947620189160, 10.20709336435470499856745072923, 11.80219272717407753281539100758, 12.18622442931276928058398029653

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