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Properties

Label 2-189-63.20-c3-0-9
Degree $2$
Conductor $189$
Sign $-0.217 - 0.976i$
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  + (4.67 + 2.70i)2-s + (10.5 + 18.3i)4-s + (−3.43 − 5.95i)5-s + (14.2 + 11.7i)7-s + 71.1i·8-s − 37.1i·10-s + (10.9 + 6.30i)11-s + (22.5 − 13.0i)13-s + (35.0 + 93.6i)14-s + (−107. + 186. i)16-s − 124.·17-s + 41.3i·19-s + (72.8 − 126. i)20-s + (34.0 + 59.0i)22-s + (97.6 − 56.4i)23-s + ⋯
L(s)  = 1  + (1.65 + 0.954i)2-s + (1.32 + 2.29i)4-s + (−0.307 − 0.532i)5-s + (0.771 + 0.635i)7-s + 3.14i·8-s − 1.17i·10-s + (0.299 + 0.172i)11-s + (0.481 − 0.277i)13-s + (0.669 + 1.78i)14-s + (−1.68 + 2.91i)16-s − 1.77·17-s + 0.498i·19-s + (0.814 − 1.41i)20-s + (0.330 + 0.571i)22-s + (0.885 − 0.511i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.76240 + 3.44610i\)
\(L(\frac12)\) \(\approx\) \(2.76240 + 3.44610i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.2 - 11.7i)T \)
good2 \( 1 + (-4.67 - 2.70i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (3.43 + 5.95i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-10.9 - 6.30i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.5 + 13.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 124.T + 4.91e3T^{2} \)
19 \( 1 - 41.3iT - 6.85e3T^{2} \)
23 \( 1 + (-97.6 + 56.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (114. + 66.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-155. + 89.5i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 + (93.6 + 162. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-99.1 + 171. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-92.1 + 159. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 359. iT - 1.48e5T^{2} \)
59 \( 1 + (-182. - 315. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (300. + 173. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (182. + 315. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 565. iT - 3.57e5T^{2} \)
73 \( 1 - 737. iT - 3.89e5T^{2} \)
79 \( 1 + (451. - 782. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (382. - 662. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 395.T + 7.04e5T^{2} \)
97 \( 1 + (243. + 140. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.63994813781359422715676566334, −11.78962252080986627682907363609, −11.02838246466934195083375229791, −8.807057230236302900859373321580, −8.139739860504342007726295929331, −6.91026186112518155582239054574, −5.89402173421932358252151345516, −4.82293372436539236327194362753, −4.06847269401438638001028073737, −2.39590176633753824970933364263, 1.38721683845619599568619775605, 2.88513799838197584056322712973, 4.09632291319286286561380198942, 4.89433845650852313216647993888, 6.33899214637067443055861883161, 7.21781228063588230928778622518, 9.082217069051338595473860042336, 10.58582433172761957538248190199, 11.18423675407056071652338050040, 11.61047357797454263388207753141

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