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Properties

Label 2-189-63.41-c1-0-3
Degree $2$
Conductor $189$
Sign $0.904 + 0.426i$
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  + (−1.02 + 0.592i)2-s + (−0.296 + 0.514i)4-s + (1.41 − 2.45i)5-s + (−0.387 − 2.61i)7-s − 3.07i·8-s + 3.36i·10-s + (−0.136 + 0.0789i)11-s + (3.41 + 1.97i)13-s + (1.95 + 2.45i)14-s + (1.23 + 2.13i)16-s + 4.14·17-s − 6.33i·19-s + (0.842 + 1.45i)20-s + (0.0935 − 0.162i)22-s + (−0.472 − 0.273i)23-s + ⋯
L(s)  = 1  + (−0.726 + 0.419i)2-s + (−0.148 + 0.257i)4-s + (0.634 − 1.09i)5-s + (−0.146 − 0.989i)7-s − 1.08i·8-s + 1.06i·10-s + (−0.0412 + 0.0237i)11-s + (0.947 + 0.546i)13-s + (0.521 + 0.656i)14-s + (0.307 + 0.532i)16-s + 1.00·17-s − 1.45i·19-s + (0.188 + 0.326i)20-s + (0.0199 − 0.0345i)22-s + (−0.0986 − 0.0569i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814337 - 0.182490i\)
\(L(\frac12)\) \(\approx\) \(0.814337 - 0.182490i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.387 + 2.61i)T \)
good2 \( 1 + (1.02 - 0.592i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.41 + 2.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.136 - 0.0789i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.41 - 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + 6.33iT - 19T^{2} \)
23 \( 1 + (0.472 + 0.273i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.112 + 0.0647i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 + (1.99 - 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.28 - 5.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.33 + 7.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.60iT - 53T^{2} \)
59 \( 1 + (-1.80 + 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.91 + 1.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + (2.16 + 3.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.22 - 5.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.05T + 89T^{2} \)
97 \( 1 + (-2.18 + 1.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.87410123426954087965499771547, −11.45254174544128942784554140816, −10.16809721927531272367349822706, −9.307027674696705628011271726600, −8.606739627649458207924499045286, −7.51665778199728448137744382952, −6.47262636646649577358634480709, −4.95960669514515008917334947596, −3.69752586245828272573291308311, −1.08014189312548087383128657237, 1.89149283792605881726851118836, 3.27288882154243200825932417482, 5.57250785424656302056517814957, 6.10874975306170576459159328006, 7.80607823382111515250238665348, 8.819660310715993346431449986650, 9.883454601855530986434059236476, 10.42864420022306771420592508745, 11.37681275996531347841757861829, 12.43933549698999179051398366543

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