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Properties

Label 2-189-63.25-c3-0-2
Degree $2$
Conductor $189$
Sign $-0.724 - 0.688i$
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  + 3.37·2-s + 3.37·4-s + (−4.87 + 8.43i)5-s + (−16.1 + 9.12i)7-s − 15.5·8-s + (−16.4 + 28.4i)10-s + (19.4 + 33.6i)11-s + (−31.2 − 54.1i)13-s + (−54.3 + 30.7i)14-s − 79.6·16-s + (−63.3 + 109. i)17-s + (22.7 + 39.3i)19-s + (−16.4 + 28.5i)20-s + (65.6 + 113. i)22-s + (76.9 − 133. i)23-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.422·4-s + (−0.435 + 0.754i)5-s + (−0.870 + 0.492i)7-s − 0.688·8-s + (−0.519 + 0.900i)10-s + (0.533 + 0.923i)11-s + (−0.667 − 1.15i)13-s + (−1.03 + 0.587i)14-s − 1.24·16-s + (−0.903 + 1.56i)17-s + (0.274 + 0.475i)19-s + (−0.184 + 0.318i)20-s + (0.636 + 1.10i)22-s + (0.697 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.724 - 0.688i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.724 - 0.688i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.530585 + 1.32832i\)
\(L(\frac12)\) \(\approx\) \(0.530585 + 1.32832i\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.1 - 9.12i)T \)
good2 \( 1 - 3.37T + 8T^{2} \)
5 \( 1 + (4.87 - 8.43i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-19.4 - 33.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (31.2 + 54.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (63.3 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22.7 - 39.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76.9 + 133. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (27.4 - 47.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 109.T + 2.97e4T^{2} \)
37 \( 1 + (-144. - 250. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-11.3 - 19.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.9 + 41.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 386.T + 1.03e5T^{2} \)
53 \( 1 + (-133. + 231. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 386.T + 2.05e5T^{2} \)
61 \( 1 - 35.6T + 2.26e5T^{2} \)
67 \( 1 - 35.7T + 3.00e5T^{2} \)
71 \( 1 - 146.T + 3.57e5T^{2} \)
73 \( 1 + (364. - 631. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 501.T + 4.93e5T^{2} \)
83 \( 1 + (169. - 293. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-104. - 180. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (56.7 - 98.2i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.68991298209165308184321896722, −11.89470486024387562238967392155, −10.65579398468529631699632606601, −9.654587905943028105661712058037, −8.382960904329254172387255081468, −6.88055990445567180542698482795, −6.16895659656463304400195120113, −4.83266739953520641288046417431, −3.63848904191684930256521006203, −2.64392094066751727773733628410, 0.41323095158487523708847794674, 2.94602571992554916509575488274, 4.14999913383806465048558412080, 4.95417907584545173544089355955, 6.31133472668640920924219063438, 7.26531220736178986837242700587, 9.031988274353763784633758613655, 9.414210605677064216722917721181, 11.33400005870063039730267853038, 11.84440193529626638357601813412

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