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Properties

Label 2-189-189.184-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.885 - 0.464i$
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  + (0.379 + 2.15i)2-s + (−0.896 − 1.48i)3-s + (−2.61 + 0.952i)4-s + (−0.347 + 1.97i)5-s + (2.85 − 2.49i)6-s + (0.0877 + 2.64i)7-s + (−0.857 − 1.48i)8-s + (−1.39 + 2.65i)9-s − 4.37·10-s + (0.213 + 1.20i)11-s + (3.75 + 3.02i)12-s + (−4.39 − 3.68i)13-s + (−5.66 + 1.19i)14-s + (3.23 − 1.25i)15-s + (−1.39 + 1.16i)16-s + 4.80·17-s + ⋯
L(s)  = 1  + (0.268 + 1.52i)2-s + (−0.517 − 0.855i)3-s + (−1.30 + 0.476i)4-s + (−0.155 + 0.881i)5-s + (1.16 − 1.01i)6-s + (0.0331 + 0.999i)7-s + (−0.303 − 0.525i)8-s + (−0.464 + 0.885i)9-s − 1.38·10-s + (0.0642 + 0.364i)11-s + (1.08 + 0.873i)12-s + (−1.21 − 1.02i)13-s + (−1.51 + 0.318i)14-s + (0.834 − 0.323i)15-s + (−0.347 + 0.291i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.885 - 0.464i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235995 + 0.958126i\)
\(L(\frac12)\) \(\approx\) \(0.235995 + 0.958126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.896 + 1.48i)T \)
7 \( 1 + (-0.0877 - 2.64i)T \)
good2 \( 1 + (-0.379 - 2.15i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.347 - 1.97i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.213 - 1.20i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (4.39 + 3.68i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 - 4.80T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + (-5.62 - 4.71i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.51 + 2.11i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-7.05 + 2.56i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.96 + 5.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.298 - 0.250i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.69 - 0.980i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.75 + 1.00i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.03 + 6.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.25 - 6.92i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.80 - 1.38i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.252 - 1.43i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.91 - 8.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.72 + 8.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.22 - 6.92i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (8.38 - 7.03i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 0.590T + 89T^{2} \)
97 \( 1 + (15.2 + 5.55i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.05372320045408157496368782421, −12.25770012762612941746901936053, −11.24057148652552405050898165856, −9.906406042045579558357102900195, −8.367415731214285060199705878992, −7.52274969964925934209775465445, −6.85585713272006368235535492576, −5.75223855961056566238415645520, −5.08073085667796896571168155577, −2.71623821707946902468083898291, 0.921764327568135049356936669301, 3.15301641343142314807152549935, 4.46995075142992267643938577495, 4.88631550902259666445971042643, 6.83388298262920459600167285454, 8.617231443097054878033804981888, 9.669300737800360144934913865428, 10.31651142263635960562226000598, 11.18877114070468738189980894441, 12.11437689587002931469602166559

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