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Properties

Label 2-189-63.5-c1-0-3
Degree $2$
Conductor $189$
Sign $0.235 + 0.971i$
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  − 1.73i·2-s − 0.999·4-s + (1.5 + 2.59i)5-s + (2 − 1.73i)7-s − 1.73i·8-s + (4.5 − 2.59i)10-s + (−1.5 − 0.866i)11-s + (1.5 + 0.866i)13-s + (−2.99 − 3.46i)14-s − 5·16-s + (1.5 + 2.59i)17-s + (−4.5 − 2.59i)19-s + (−1.49 − 2.59i)20-s + (−1.49 + 2.59i)22-s + (−4.5 + 2.59i)23-s + ⋯
L(s)  = 1  − 1.22i·2-s − 0.499·4-s + (0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s − 0.612i·8-s + (1.42 − 0.821i)10-s + (−0.452 − 0.261i)11-s + (0.416 + 0.240i)13-s + (−0.801 − 0.925i)14-s − 1.25·16-s + (0.363 + 0.630i)17-s + (−1.03 − 0.596i)19-s + (−0.335 − 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.938 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.235 + 0.971i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09372 - 0.860197i\)
\(L(\frac12)\) \(\approx\) \(1.09372 - 0.860197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.11376097525866690997451635786, −11.13167810783007110224673052767, −10.53119838694185963234949928565, −10.00279037089535379135670466250, −8.487508167254132979072366112366, −7.12425691836572134446665467770, −6.10379108144574420531505957857, −4.29954279178598716552846928541, −3.00861810588390214752082583317, −1.75705901659726654071396250373, 2.05497299292172316669943338762, 4.68673761222492747515077923309, 5.46118461214875833774081558645, 6.33134503448031498332437570336, 7.88773791311141527155313763300, 8.444276237637559525034583011494, 9.396968915543597441072449387229, 10.76065097039486489265060144778, 12.04191054325712859493301479954, 12.86706278285562702997062033078

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