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Properties

Label 2-189-63.4-c3-0-4
Degree $2$
Conductor $189$
Sign $0.359 - 0.933i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.83 − 3.17i)2-s + (−2.73 − 4.73i)4-s − 14.7·5-s + (0.242 + 18.5i)7-s + 9.29·8-s + (−27.1 + 46.9i)10-s − 48.5·11-s + (−33.6 + 58.1i)13-s + (59.2 + 33.2i)14-s + (38.9 − 67.4i)16-s + (5.40 − 9.36i)17-s + (67.2 + 116. i)19-s + (40.3 + 69.9i)20-s + (−89.1 + 154. i)22-s − 84.3·23-s + ⋯
L(s)  = 1  + (0.648 − 1.12i)2-s + (−0.341 − 0.591i)4-s − 1.32·5-s + (0.0130 + 0.999i)7-s + 0.410·8-s + (−0.857 + 1.48i)10-s − 1.33·11-s + (−0.716 + 1.24i)13-s + (1.13 + 0.633i)14-s + (0.608 − 1.05i)16-s + (0.0771 − 0.133i)17-s + (0.811 + 1.40i)19-s + (0.451 + 0.782i)20-s + (−0.864 + 1.49i)22-s − 0.764·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.741234 + 0.508605i\)
\(L(\frac12)\) \(\approx\) \(0.741234 + 0.508605i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.242 - 18.5i)T \)
good2 \( 1 + (-1.83 + 3.17i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 14.7T + 125T^{2} \)
11 \( 1 + 48.5T + 1.33e3T^{2} \)
13 \( 1 + (33.6 - 58.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-5.40 + 9.36i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-67.2 - 116. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 84.3T + 1.21e4T^{2} \)
29 \( 1 + (-55.1 - 95.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (75.5 + 130. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (152. + 263. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (127. - 220. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-41.3 - 71.5i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-23.0 + 39.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-3.20 + 5.55i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-5.59 - 9.69i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (136. - 235. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-28.7 - 49.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 521.T + 3.57e5T^{2} \)
73 \( 1 + (189. - 327. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-472. + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-411. - 711. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-12.4 - 21.6i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-22.7 - 39.3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.00553526231013914566268292478, −11.77659794035458822684397977075, −10.67043052337721405200312820329, −9.592435857742348332680863026323, −8.147502708797282280731193386616, −7.39193976665583107658557379867, −5.48983380408873147817838621390, −4.39827629819622861958500357349, −3.28389589322139521663268382562, −2.06432826268115925858909061781, 0.30862868955953955943204841058, 3.25236443903682542885557064531, 4.55982299113369948198417167970, 5.34697960759794133493474757103, 6.96140963740026241867493849898, 7.63320028979481121205660147143, 8.183422489213655585974003031128, 10.17396579088773323349091022525, 10.87494546529995123676937347550, 12.15189290561732368392763768176

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