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Properties

Label 2-189-63.31-c2-0-4
Degree $2$
Conductor $189$
Sign $-0.363 - 0.931i$
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  + (−1.91 + 3.32i)2-s + (−5.35 − 9.27i)4-s + 0.216i·5-s + (6.73 − 1.90i)7-s + 25.7·8-s + (−0.719 − 0.415i)10-s − 4.44·11-s + (14.9 + 8.61i)13-s + (−6.57 + 26.0i)14-s + (−27.9 + 48.4i)16-s + (4.66 + 2.69i)17-s + (−6.88 + 3.97i)19-s + (2.00 − 1.16i)20-s + (8.53 − 14.7i)22-s + 19.0·23-s + ⋯
L(s)  = 1  + (−0.959 + 1.66i)2-s + (−1.33 − 2.31i)4-s + 0.0433i·5-s + (0.962 − 0.272i)7-s + 3.21·8-s + (−0.0719 − 0.0415i)10-s − 0.404·11-s + (1.14 + 0.663i)13-s + (−0.469 + 1.85i)14-s + (−1.74 + 3.02i)16-s + (0.274 + 0.158i)17-s + (−0.362 + 0.209i)19-s + (0.100 − 0.0580i)20-s + (0.387 − 0.671i)22-s + 0.828·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.363 - 0.931i$
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: \((2,\ 189,\ (\ :1),\ -0.363 - 0.931i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.538311 + 0.788244i\)
\(L(\frac12)\) \(\approx\) \(0.538311 + 0.788244i\)
\(L(2)\) 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 \( 1 + (-6.73 + 1.90i)T \)
good2 \( 1 + (1.91 - 3.32i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 - 0.216iT - 25T^{2} \)
11 \( 1 + 4.44T + 121T^{2} \)
13 \( 1 + (-14.9 - 8.61i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-4.66 - 2.69i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.88 - 3.97i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 19.0T + 529T^{2} \)
29 \( 1 + (3.57 + 6.18i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (20.0 - 11.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.16 - 8.95i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (3.32 + 1.91i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.1 - 52.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42.4 - 24.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (25.0 - 43.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-75.2 + 43.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.6 - 20.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (32.1 + 55.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 11.8T + 5.04e3T^{2} \)
73 \( 1 + (27.7 + 16.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (7.26 - 12.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (94.9 - 54.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-111. + 64.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-63.3 + 36.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.03737964459669562501594418234, −11.15926620193752938271381140147, −10.49372223912298572469708221175, −9.189182375141332379446397345694, −8.457736209696979112286058369735, −7.59832877419535868065720665958, −6.61898253190520418404719784588, −5.53537616499996524256703045389, −4.42636852745520603625566444413, −1.26032076933883662563523156510, 1.00872823913251800788531875330, 2.46130977825031372588129160336, 3.79422955382307175879692923811, 5.21216260365451057514072027990, 7.43663302145840681052066679247, 8.481844241301044746877788631668, 9.003499431923671385958517286120, 10.40047136969364753370834690113, 10.92625361978949101231617782117, 11.71739235387677448041221651163

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