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Properties

Label 2-189-63.31-c2-0-7
Degree $2$
Conductor $189$
Sign $0.927 + 0.373i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 2.30i)2-s + (−1.52 − 2.64i)4-s + 9.20i·5-s + (6.96 + 0.687i)7-s + 2.50·8-s + (21.1 + 12.2i)10-s − 7.05·11-s + (4.15 + 2.40i)13-s + (10.8 − 15.1i)14-s + (9.43 − 16.3i)16-s + (2.74 + 1.58i)17-s + (1.70 − 0.986i)19-s + (24.3 − 14.0i)20-s + (−9.37 + 16.2i)22-s + 5.02·23-s + ⋯
L(s)  = 1  + (0.664 − 1.15i)2-s + (−0.382 − 0.662i)4-s + 1.84i·5-s + (0.995 + 0.0981i)7-s + 0.312·8-s + (2.11 + 1.22i)10-s − 0.641·11-s + (0.319 + 0.184i)13-s + (0.773 − 1.07i)14-s + (0.589 − 1.02i)16-s + (0.161 + 0.0931i)17-s + (0.0899 − 0.0519i)19-s + (1.21 − 0.703i)20-s + (−0.425 + 0.737i)22-s + 0.218·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.927 + 0.373i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.927 + 0.373i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.29329 - 0.444904i\)
\(L(\frac12)\) \(\approx\) \(2.29329 - 0.444904i\)
\(L(2)\) 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 \( 1 + (-6.96 - 0.687i)T \)
good2 \( 1 + (-1.32 + 2.30i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 - 9.20iT - 25T^{2} \)
11 \( 1 + 7.05T + 121T^{2} \)
13 \( 1 + (-4.15 - 2.40i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-2.74 - 1.58i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.70 + 0.986i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 5.02T + 529T^{2} \)
29 \( 1 + (22.9 + 39.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.2 + 7.63i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-17.6 - 30.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (48.4 + 27.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (3.45 + 5.97i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-44.4 - 25.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-10.3 + 17.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-42.7 + 24.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.99 - 1.15i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.4 + 26.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 81.2T + 5.04e3T^{2} \)
73 \( 1 + (61.7 + 35.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-14.5 + 25.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (55.9 - 32.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (89.0 - 51.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (48.7 - 28.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.91288277035849095077458146769, −11.32430478137911003750643222980, −10.66265635344915682062572968909, −9.914605539850049499639093848445, −8.042311827556648519772412293336, −7.14188569470765516711192485124, −5.70861620193292603466211997466, −4.24042615050534827096910562104, −3.04590784446604674983442023402, −2.05685305122644140294733821253, 1.37702147284223447030722129631, 4.19700513694869181013316665225, 5.11471573280542884228719549921, 5.63209121522810088945976556903, 7.32242447572879134088128840472, 8.200250454131655860670400055989, 8.900082115796944378905655530135, 10.42229689758396509707488648290, 11.70105280021036138772938525870, 12.81435425041426471380053114037

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