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Properties

Label 2-189-7.4-c3-0-21
Degree $2$
Conductor $189$
Sign $0.868 + 0.496i$
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  + (0.720 + 1.24i)2-s + (2.96 − 5.12i)4-s + (3.30 + 5.71i)5-s + (−10.3 − 15.3i)7-s + 20.0·8-s + (−4.76 + 8.24i)10-s + (14.5 − 25.2i)11-s + 16.7·13-s + (11.7 − 23.9i)14-s + (−9.21 − 15.9i)16-s + (23.5 − 40.7i)17-s + (−1.71 − 2.97i)19-s + 39.0·20-s + 42.0·22-s + (−14.7 − 25.5i)23-s + ⋯
L(s)  = 1  + (0.254 + 0.441i)2-s + (0.370 − 0.640i)4-s + (0.295 + 0.511i)5-s + (−0.557 − 0.830i)7-s + 0.887·8-s + (−0.150 + 0.260i)10-s + (0.399 − 0.692i)11-s + 0.358·13-s + (0.224 − 0.457i)14-s + (−0.143 − 0.249i)16-s + (0.336 − 0.582i)17-s + (−0.0207 − 0.0358i)19-s + 0.437·20-s + 0.407·22-s + (−0.133 − 0.231i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.868 + 0.496i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.17834 - 0.579007i\)
\(L(\frac12)\) \(\approx\) \(2.17834 - 0.579007i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.3 + 15.3i)T \)
good2 \( 1 + (-0.720 - 1.24i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-3.30 - 5.71i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-14.5 + 25.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 16.7T + 2.19e3T^{2} \)
17 \( 1 + (-23.5 + 40.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1.71 + 2.97i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (14.7 + 25.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 223.T + 2.43e4T^{2} \)
31 \( 1 + (60.3 - 104. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (17.4 + 30.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 192.T + 6.89e4T^{2} \)
43 \( 1 + 5.61T + 7.95e4T^{2} \)
47 \( 1 + (-179. - 311. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (16.6 - 28.8i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (371. - 642. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (329. + 570. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (470. - 815. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 871.T + 3.57e5T^{2} \)
73 \( 1 + (-366. + 634. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (671. + 1.16e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 588.T + 5.71e5T^{2} \)
89 \( 1 + (-623. - 1.08e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 691.T + 9.12e5T^{2} \)
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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

−11.97800103358481980175745686881, −10.73885002900419864883121646606, −10.33833784705689080251722226550, −9.119957595013647755909740152561, −7.60114373634617714771181527346, −6.61992959108208189934151661955, −5.97197593200055355605303962505, −4.49088899808930372812078476146, −2.94882873696551600990685333404, −1.00726866050985513011172058623, 1.71864709085955492239094646098, 3.07803439195255393556716710340, 4.38487754429386478855983560750, 5.80486365166671159107925388233, 6.97940494684330358792375670861, 8.250333342638481778838835748175, 9.203974562690957666985233391497, 10.26861481255553736721064619953, 11.47429001795464660900243950591, 12.35612231912170113953198643694

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