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Properties

Label 2-189-63.38-c3-0-16
Degree $2$
Conductor $189$
Sign $0.680 + 0.732i$
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  − 0.747i·2-s + 7.44·4-s + (−4.35 + 7.53i)5-s + (−11.6 − 14.4i)7-s − 11.5i·8-s + (5.63 + 3.25i)10-s + (35.9 − 20.7i)11-s + (33.0 − 19.0i)13-s + (−10.7 + 8.68i)14-s + 50.8·16-s + (39.9 − 69.2i)17-s + (−49.4 + 28.5i)19-s + (−32.3 + 56.0i)20-s + (−15.5 − 26.8i)22-s + (153. + 88.5i)23-s + ⋯
L(s)  = 1  − 0.264i·2-s + 0.930·4-s + (−0.389 + 0.674i)5-s + (−0.627 − 0.778i)7-s − 0.510i·8-s + (0.178 + 0.102i)10-s + (0.985 − 0.568i)11-s + (0.704 − 0.406i)13-s + (−0.205 + 0.165i)14-s + 0.795·16-s + (0.570 − 0.987i)17-s + (−0.596 + 0.344i)19-s + (−0.361 + 0.626i)20-s + (−0.150 − 0.260i)22-s + (1.38 + 0.802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.680 + 0.732i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.680 + 0.732i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.86084 - 0.811779i\)
\(L(\frac12)\) \(\approx\) \(1.86084 - 0.811779i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.6 + 14.4i)T \)
good2 \( 1 + 0.747iT - 8T^{2} \)
5 \( 1 + (4.35 - 7.53i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.9 + 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.0 + 19.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-39.9 + 69.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.4 - 28.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-153. - 88.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-60.1 - 34.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 260. iT - 2.97e4T^{2} \)
37 \( 1 + (74.9 + 129. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (54.7 + 94.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 295.T + 1.03e5T^{2} \)
53 \( 1 + (-263. - 152. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 720.T + 2.05e5T^{2} \)
61 \( 1 - 648. iT - 2.26e5T^{2} \)
67 \( 1 + 194.T + 3.00e5T^{2} \)
71 \( 1 - 98.7iT - 3.57e5T^{2} \)
73 \( 1 + (-226. - 130. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + (285. - 494. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-89.5 - 155. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-956. - 552. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.70018858813526593916911926790, −11.07197896529460826189615530727, −10.29456162284532627912045431657, −9.119187232319703690286090105563, −7.54908888406010909380084550342, −6.89223600776968996566266907681, −5.89010739876244493499069358008, −3.77895763898094680248402841187, −3.02911965056962965167033667342, −1.02442286875705981399440844373, 1.50980072619075947817261592919, 3.17229868392784134388909339126, 4.72090926641037491834798329763, 6.21737132649188902396095090448, 6.79926495423399781472683291546, 8.325151459360393462086588055884, 8.980747200875898831076422470696, 10.33161371623788552276976450228, 11.42364549375028554335086154858, 12.31805778555311909315823011876

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