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Properties

Label 2-189-63.2-c2-0-11
Degree $2$
Conductor $189$
Sign $0.292 + 0.956i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.19 − 1.84i)2-s + (4.81 − 8.33i)4-s + 5.83i·5-s + (3.24 − 6.20i)7-s − 20.7i·8-s + (10.7 + 18.6i)10-s − 3.36i·11-s + (0.158 + 0.273i)13-s + (−1.05 − 25.8i)14-s + (−19.0 − 32.9i)16-s + (−2.99 + 1.73i)17-s + (−12.0 + 20.8i)19-s + (48.5 + 28.0i)20-s + (−6.20 − 10.7i)22-s + 34.0i·23-s + ⋯
L(s)  = 1  + (1.59 − 0.922i)2-s + (1.20 − 2.08i)4-s + 1.16i·5-s + (0.464 − 0.885i)7-s − 2.59i·8-s + (1.07 + 1.86i)10-s − 0.305i·11-s + (0.0121 + 0.0210i)13-s + (−0.0753 − 1.84i)14-s + (−1.19 − 2.06i)16-s + (−0.176 + 0.101i)17-s + (−0.633 + 1.09i)19-s + (2.42 + 1.40i)20-s + (−0.282 − 0.488i)22-s + 1.47i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.86898 - 2.12293i\)
\(L(\frac12)\) \(\approx\) \(2.86898 - 2.12293i\)
\(L(2)\) 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 \( 1 + (-3.24 + 6.20i)T \)
good2 \( 1 + (-3.19 + 1.84i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 5.83iT - 25T^{2} \)
11 \( 1 + 3.36iT - 121T^{2} \)
13 \( 1 + (-0.158 - 0.273i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (2.99 - 1.73i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (12.0 - 20.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 34.0iT - 529T^{2} \)
29 \( 1 + (31.6 + 18.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (12.0 - 20.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.23 + 7.33i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-39.9 + 23.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (26.4 - 45.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-17.2 + 9.97i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-29.2 + 16.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (31.5 + 18.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.02 + 10.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-17.4 + 30.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 85.7iT - 5.04e3T^{2} \)
73 \( 1 + (35.3 + 61.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-50.7 - 87.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (26.0 + 15.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-46.5 - 26.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (3.60 - 6.25i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.14045090030130543482908836079, −11.06463508023528336293011600160, −10.81276680054674897441197872137, −9.765386451926683126020923939042, −7.66320637021757655801008920396, −6.55900661182034923782816776987, −5.52672054555962989255608161542, −4.10075296987007868428138913660, −3.30651273709566489922553200554, −1.80700259208736682610253802964, 2.45857769268568513789882878598, 4.27935815920362408378519400470, 4.99372031538301843172249708510, 5.89684287126375407096304003167, 7.09980215855117899739519594300, 8.324133679691532891609044198959, 9.059229444105964631191877225231, 11.09626402159534660401701967319, 12.10920344627448114455154775409, 12.74532373023913232096211500986

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