Properties

Label 2-189-189.4-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.707 + 0.706i$
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  + (0.373 + 2.12i)2-s + (−1.62 + 0.609i)3-s + (−2.47 + 0.901i)4-s + (−1.18 + 0.430i)5-s + (−1.89 − 3.21i)6-s + (−2.62 − 0.302i)7-s + (−0.686 − 1.18i)8-s + (2.25 − 1.97i)9-s + (−1.35 − 2.34i)10-s + (0.527 + 0.192i)11-s + (3.46 − 2.97i)12-s + (−3.76 + 1.37i)13-s + (−0.341 − 5.68i)14-s + (1.65 − 1.41i)15-s + (−1.77 + 1.49i)16-s + (1.82 + 3.15i)17-s + ⋯
L(s)  = 1  + (0.264 + 1.49i)2-s + (−0.936 + 0.351i)3-s + (−1.23 + 0.450i)4-s + (−0.528 + 0.192i)5-s + (−0.775 − 1.31i)6-s + (−0.993 − 0.114i)7-s + (−0.242 − 0.420i)8-s + (0.752 − 0.658i)9-s + (−0.428 − 0.741i)10-s + (0.159 + 0.0579i)11-s + (1.00 − 0.858i)12-s + (−1.04 + 0.380i)13-s + (−0.0911 − 1.51i)14-s + (0.427 − 0.366i)15-s + (−0.444 + 0.372i)16-s + (0.442 + 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.707 + 0.706i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.214386 - 0.518234i\)
\(L(\frac12)\) \(\approx\) \(0.214386 - 0.518234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.609i)T \)
7 \( 1 + (2.62 + 0.302i)T \)
good2 \( 1 + (-0.373 - 2.12i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (1.18 - 0.430i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.527 - 0.192i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.76 - 1.37i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.82 - 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 + 2.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.669 - 3.79i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-7.38 - 2.68i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (7.43 - 2.70i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 - 6.30T + 37T^{2} \)
41 \( 1 + (-2.95 + 1.07i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.190 - 1.08i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.72 + 2.80i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.59 - 4.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.22 - 5.22i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-9.96 - 3.62i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.78 + 10.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.01 - 6.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.24T + 73T^{2} \)
79 \( 1 + (2.46 + 13.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-13.4 - 4.88i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-3.34 + 5.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.378 + 2.14i)T + (-91.1 + 33.1i)T^{2} \)
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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

−13.23481427688456677391166106023, −12.33109856486091722789169385265, −11.31332065743025112858809686713, −10.08065217028448455811663264154, −9.127550128973111683304965022199, −7.58033365486429965660910893939, −6.89562449809008178723631466434, −5.97908928605041412829825947930, −4.93680564145662324235998911301, −3.74283150121321996592088315612, 0.50779818724924139203449371537, 2.55216478154649171343863085233, 3.99906253034199693121730493051, 5.18705687739402893332130917511, 6.60064867606401150981049542766, 7.82064562366709118915942722120, 9.634720011782782821700291465223, 10.11297253505091391610028271089, 11.27222764935527449769584111152, 12.03789611642410938024639073936

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