↑

Properties

Label 2-189-63.58-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.391 - 0.920i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.670·2-s − 1.55·4-s + (0.712 + 1.23i)5-s + (−2.36 + 1.19i)7-s + 2.38·8-s + (−0.477 − 0.827i)10-s + (−2.46 + 4.27i)11-s + (−1.37 + 2.38i)13-s + (1.58 − 0.801i)14-s + 1.50·16-s + (−0.559 − 0.969i)17-s + (−2.00 + 3.47i)19-s + (−1.10 − 1.91i)20-s + (1.65 − 2.86i)22-s + (2.71 + 4.70i)23-s + ⋯
L(s)  = 1  − 0.473·2-s − 0.775·4-s + (0.318 + 0.551i)5-s + (−0.892 + 0.451i)7-s + 0.841·8-s + (−0.151 − 0.261i)10-s + (−0.743 + 1.28i)11-s + (−0.381 + 0.661i)13-s + (0.422 − 0.214i)14-s + 0.376·16-s + (−0.135 − 0.235i)17-s + (−0.460 + 0.797i)19-s + (−0.247 − 0.427i)20-s + (0.352 − 0.610i)22-s + (0.566 + 0.981i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300771 + 0.455011i\)
\(L(\frac12)\) \(\approx\) \(0.300771 + 0.455011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 1.19i)T \)
good2 \( 1 + 0.670T + 2T^{2} \)
5 \( 1 + (-0.712 - 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.46 - 4.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.559 + 0.969i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.00 - 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.71 - 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.47T + 47T^{2} \)
53 \( 1 + (-0.410 - 0.710i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.58T + 59T^{2} \)
61 \( 1 - 0.0752T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.91679638534493224966855603077, −12.02854457820974600888364751308, −10.50895132357294926084559613822, −9.805636225465597624338679940722, −9.168018454379925423402246052823, −7.82352487517781937212328617230, −6.81465657693363780074709019384, −5.45583989779084159023499424549, −4.12623526998851059966908405149, −2.36233235253502520283381866336, 0.57361318231370955833291820676, 3.15914509184224996445722474350, 4.71051099598126858416115818280, 5.80411674563046393058964950977, 7.27333836697874264002175177003, 8.529232073940013129368371568443, 9.102324681291407613922754041785, 10.26313238156927473884741277279, 10.87978007596253291150537572072, 12.69529252420163054082921334770

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