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Properties

Label 2-189-63.41-c3-0-16
Degree $2$
Conductor $189$
Sign $-0.250 + 0.968i$
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  + (−2.59 + 1.49i)2-s + (0.478 − 0.828i)4-s + (7.80 − 13.5i)5-s + (17.7 − 5.41i)7-s − 21.0i·8-s + 46.6i·10-s + (−46.4 + 26.8i)11-s + (−29.1 − 16.8i)13-s + (−37.8 + 40.5i)14-s + (35.3 + 61.2i)16-s − 43.5·17-s − 32.2i·19-s + (−7.46 − 12.9i)20-s + (80.2 − 139. i)22-s + (−129. − 74.6i)23-s + ⋯
L(s)  = 1  + (−0.916 + 0.529i)2-s + (0.0598 − 0.103i)4-s + (0.697 − 1.20i)5-s + (0.956 − 0.292i)7-s − 0.931i·8-s + 1.47i·10-s + (−1.27 + 0.735i)11-s + (−0.621 − 0.358i)13-s + (−0.721 + 0.773i)14-s + (0.552 + 0.957i)16-s − 0.621·17-s − 0.389i·19-s + (−0.0834 − 0.144i)20-s + (0.778 − 1.34i)22-s + (−1.17 − 0.676i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.250 + 0.968i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.378316 - 0.488833i\)
\(L(\frac12)\) \(\approx\) \(0.378316 - 0.488833i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.7 + 5.41i)T \)
good2 \( 1 + (2.59 - 1.49i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-7.80 + 13.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (46.4 - 26.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (29.1 + 16.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 43.5T + 4.91e3T^{2} \)
19 \( 1 + 32.2iT - 6.85e3T^{2} \)
23 \( 1 + (129. + 74.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-127. + 73.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (61.1 + 35.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 40.3T + 5.06e4T^{2} \)
41 \( 1 + (179. - 310. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (253. + 439. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (228. + 395. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 213. iT - 1.48e5T^{2} \)
59 \( 1 + (-159. + 276. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-303. + 175. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (289. - 501. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 787. iT - 3.57e5T^{2} \)
73 \( 1 + 146. iT - 3.89e5T^{2} \)
79 \( 1 + (193. + 334. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-98.9 - 171. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 596.T + 7.04e5T^{2} \)
97 \( 1 + (-631. + 364. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.92848509119549981116049320628, −10.32717261628591275558774221045, −9.781539103094506693544781666765, −8.516229605403726202949352354750, −8.085767841205867427355019245635, −6.93018585612691228275080406298, −5.31322880779416271326793839330, −4.48062172600994191117399934402, −2.01998056158706205497011098599, −0.35226889107010617392814313002, 1.84059595005664743290032283500, 2.79696476660802164328239460373, 5.02743564940623056115164144343, 6.08599817032120945534828231060, 7.59691204027480848971734584820, 8.455955934229036517120821652181, 9.645286069424727573637898880134, 10.49498991212003704162559399553, 11.00687615945196890364502519849, 11.96009089508251143283713802763

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