Properties

Label 2-189-63.25-c3-0-7
Degree $2$
Conductor $189$
Sign $0.240 - 0.970i$
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  + 1.80·2-s − 4.72·4-s + (−1.04 + 1.81i)5-s + (18.3 + 2.47i)7-s − 23.0·8-s + (−1.89 + 3.28i)10-s + (11.7 + 20.3i)11-s + (27.8 + 48.2i)13-s + (33.2 + 4.47i)14-s − 3.83·16-s + (−55.3 + 95.9i)17-s + (9.75 + 16.8i)19-s + (4.95 − 8.58i)20-s + (21.2 + 36.7i)22-s + (−6.87 + 11.9i)23-s + ⋯
L(s)  = 1  + 0.639·2-s − 0.590·4-s + (−0.0938 + 0.162i)5-s + (0.991 + 0.133i)7-s − 1.01·8-s + (−0.0600 + 0.103i)10-s + (0.321 + 0.557i)11-s + (0.594 + 1.02i)13-s + (0.633 + 0.0854i)14-s − 0.0599·16-s + (−0.790 + 1.36i)17-s + (0.117 + 0.203i)19-s + (0.0554 − 0.0960i)20-s + (0.205 + 0.356i)22-s + (−0.0623 + 0.107i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.240 - 0.970i$
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: \((2,\ 189,\ (\ :3/2),\ 0.240 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47209 + 1.15242i\)
\(L(\frac12)\) \(\approx\) \(1.47209 + 1.15242i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 - 2.47i)T \)
good2 \( 1 - 1.80T + 8T^{2} \)
5 \( 1 + (1.04 - 1.81i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-11.7 - 20.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-27.8 - 48.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (55.3 - 95.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.75 - 16.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.87 - 11.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (59.9 - 103. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 158.T + 2.97e4T^{2} \)
37 \( 1 + (79.4 + 137. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (208. + 361. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-131. + 227. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 190.T + 1.03e5T^{2} \)
53 \( 1 + (175. - 304. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 127.T + 2.05e5T^{2} \)
61 \( 1 - 724.T + 2.26e5T^{2} \)
67 \( 1 + 998.T + 3.00e5T^{2} \)
71 \( 1 - 404.T + 3.57e5T^{2} \)
73 \( 1 + (120. - 208. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 921.T + 4.93e5T^{2} \)
83 \( 1 + (-502. + 869. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-239. - 414. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-431. + 747. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.37799573038711145600224038525, −11.52198506149737006412725762483, −10.51182911930871826540576761445, −9.086081077280104986883288424713, −8.486936451674154782417600273625, −7.02309155816194220981336326404, −5.77905479136525683688763779950, −4.60114870790275812461667088461, −3.75533225653161792477960069290, −1.75274961016373985176857385239, 0.73144301628962538661266506922, 2.97908725320663553029458366857, 4.40054759048706036428965974317, 5.18933928791661629701077973623, 6.42498560213409856586896270040, 8.039931495812313924870450415810, 8.705405403770534163249145113246, 9.905708503088853966552297940463, 11.21865208655221579701426792403, 11.88833115028068099009390111527

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