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Properties

Label 2-189-63.41-c3-0-14
Degree $2$
Conductor $189$
Sign $0.998 + 0.0523i$
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  + (1.10 − 0.638i)2-s + (−3.18 + 5.51i)4-s + (1.59 − 2.76i)5-s + (−4.68 − 17.9i)7-s + 18.3i·8-s − 4.07i·10-s + (38.0 − 21.9i)11-s + (65.6 + 37.8i)13-s + (−16.6 − 16.8i)14-s + (−13.7 − 23.8i)16-s + 104.·17-s + 74.7i·19-s + (10.1 + 17.5i)20-s + (28.0 − 48.6i)22-s + (−46.6 − 26.9i)23-s + ⋯
L(s)  = 1  + (0.390 − 0.225i)2-s + (−0.398 + 0.689i)4-s + (0.142 − 0.247i)5-s + (−0.252 − 0.967i)7-s + 0.810i·8-s − 0.128i·10-s + (1.04 − 0.602i)11-s + (1.39 + 0.808i)13-s + (−0.317 − 0.321i)14-s + (−0.215 − 0.372i)16-s + 1.49·17-s + 0.902i·19-s + (0.113 + 0.196i)20-s + (0.272 − 0.471i)22-s + (−0.422 − 0.244i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.998 + 0.0523i$
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: \((2,\ 189,\ (\ :3/2),\ 0.998 + 0.0523i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.13786 - 0.0560221i\)
\(L(\frac12)\) \(\approx\) \(2.13786 - 0.0560221i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.68 + 17.9i)T \)
good2 \( 1 + (-1.10 + 0.638i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-1.59 + 2.76i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-38.0 + 21.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-65.6 - 37.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 - 74.7iT - 6.85e3T^{2} \)
23 \( 1 + (46.6 + 26.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-26.3 + 15.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-111. - 64.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 46.1T + 5.06e4T^{2} \)
41 \( 1 + (-213. + 369. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (166. + 287. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (170. + 294. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 235. iT - 1.48e5T^{2} \)
59 \( 1 + (272. - 471. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-321. + 185. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (53.3 - 92.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 974. iT - 3.57e5T^{2} \)
73 \( 1 - 576. iT - 3.89e5T^{2} \)
79 \( 1 + (10.6 + 18.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (168. + 292. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (63.5 - 36.6i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.12880399223506652032644235379, −11.36304455961833949639214761265, −10.17708078543887816294971874844, −8.981132875356868722573921709294, −8.169332501891337780652232893921, −6.89534010419395888082606698451, −5.63776490332394275843162108590, −4.02308999039837476726641942259, −3.51880115199828467659748943512, −1.22145528187796604592217348689, 1.19230898345011976621517070015, 3.18660907930843934282858915037, 4.63799017544055023761258109972, 5.91897431943201111785684075933, 6.44056957859475215516763458189, 8.122101823838661908840859480339, 9.291536745215382447056198779325, 9.963491649851374117846015225830, 11.16235121851584627477675352384, 12.28961122730110529363231332711

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