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Properties

Label 2-189-63.5-c1-0-5
Degree $2$
Conductor $189$
Sign $-0.846 - 0.532i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.59i·2-s − 4.72·4-s + (−0.626 − 1.08i)5-s + (−1.89 − 1.84i)7-s + 7.07i·8-s + (−2.81 + 1.62i)10-s + (0.534 + 0.308i)11-s + (1.06 + 0.613i)13-s + (−4.78 + 4.92i)14-s + 8.88·16-s + (−2.21 − 3.83i)17-s + (−1.64 − 0.950i)19-s + (2.96 + 5.12i)20-s + (0.799 − 1.38i)22-s + (4.11 − 2.37i)23-s + ⋯
L(s)  = 1  − 1.83i·2-s − 2.36·4-s + (−0.280 − 0.485i)5-s + (−0.717 − 0.696i)7-s + 2.50i·8-s + (−0.889 + 0.513i)10-s + (0.161 + 0.0929i)11-s + (0.294 + 0.170i)13-s + (−1.27 + 1.31i)14-s + 2.22·16-s + (−0.537 − 0.930i)17-s + (−0.377 − 0.218i)19-s + (0.662 + 1.14i)20-s + (0.170 − 0.295i)22-s + (0.857 − 0.495i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.846 - 0.532i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216933 + 0.751445i\)
\(L(\frac12)\) \(\approx\) \(0.216933 + 0.751445i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 + 1.84i)T \)
good2 \( 1 + 2.59iT - 2T^{2} \)
5 \( 1 + (0.626 + 1.08i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.534 - 0.308i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.06 - 0.613i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.21 + 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.64 + 0.950i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 + 2.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.07 + 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.48iT - 31T^{2} \)
37 \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.09 + 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.61T + 47T^{2} \)
53 \( 1 + (-2.67 + 1.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.56T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-9.95 + 5.74i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.03T + 79T^{2} \)
83 \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.811 - 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.76 + 5.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.96509003471453271374809124560, −11.03559111847577953556618441914, −10.22971625882403681276670603545, −9.290853638308887486612146086714, −8.477477657412254198536326977576, −6.79622897816441213761830680049, −4.84849055104940049734896436951, −3.95377053494911115141518452003, −2.65888259288660502433495993037, −0.73405529705037242408714359785, 3.46497479826553202915558422995, 4.95897476055166927272855153680, 6.19584429311653994373531911368, 6.72426142450614828780832313497, 7.971158998961373518773675310685, 8.801584361938396287794126382598, 9.730461107392377836486187261787, 11.11160489655364383867128328594, 12.65508395984390644313190799760, 13.30821700522437731150432931346

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