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Properties

Label 2-189-7.4-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.811 - 0.583i$
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  + (1.34 + 2.33i)2-s + (−2.64 + 4.57i)4-s + (−0.794 − 1.37i)5-s + (1.23 + 2.33i)7-s − 8.87·8-s + (2.14 − 3.71i)10-s + (0.150 − 0.260i)11-s + 2.81·13-s + (−3.79 + 6.05i)14-s + (−6.69 − 11.5i)16-s + (2.93 − 5.08i)17-s + (1.14 + 1.98i)19-s + 8.39·20-s + 0.810·22-s + (0.944 + 1.63i)23-s + ⋯
L(s)  = 1  + (0.954 + 1.65i)2-s + (−1.32 + 2.28i)4-s + (−0.355 − 0.615i)5-s + (0.468 + 0.883i)7-s − 3.13·8-s + (0.677 − 1.17i)10-s + (0.0452 − 0.0784i)11-s + 0.779·13-s + (−1.01 + 1.61i)14-s + (−1.67 − 2.89i)16-s + (0.712 − 1.23i)17-s + (0.262 + 0.454i)19-s + 1.87·20-s + 0.172·22-s + (0.196 + 0.341i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.811 - 0.583i$
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: \((2,\ 189,\ (\ :1/2),\ -0.811 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513720 + 1.59439i\)
\(L(\frac12)\) \(\approx\) \(0.513720 + 1.59439i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 2.33i)T \)
good2 \( 1 + (-1.34 - 2.33i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.150 + 0.260i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 + (-2.93 + 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.944 - 1.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.52T + 29T^{2} \)
31 \( 1 + (-2.40 + 4.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.90T + 41T^{2} \)
43 \( 1 + 9.09T + 43T^{2} \)
47 \( 1 + (1.60 + 2.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.00 + 1.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.44 - 4.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.78 + 6.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.356 - 0.616i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + (-5.83 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.833 + 1.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.43T + 83T^{2} \)
89 \( 1 + (4.67 + 8.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−13.27307347723461686542889933820, −12.18835223749911752231583158147, −11.64102447129663135611859207618, −9.448198817602835847915901480729, −8.433852872728916520660334155356, −7.86419658766660756451435939706, −6.57025391529394888713174948974, −5.47851722756754097897830001174, −4.76684561509377538972656150986, −3.36939575975004998159327043845, 1.44210283564697640358080730706, 3.23232935400843525132708911932, 4.04102450783686135612168721122, 5.28103108177296070175151546898, 6.69762645908457850119297206861, 8.333098547836233402548535942330, 9.768227130883434834230553096197, 10.72392550859315756278631569045, 11.08049947778631818627689567547, 12.10021605066844905513002647246

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