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Properties

Label 2-189-63.25-c1-0-4
Degree $2$
Conductor $189$
Sign $0.997 + 0.0717i$
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  + 1.84·2-s + 1.39·4-s + (0.667 − 1.15i)5-s + (1.90 + 1.83i)7-s − 1.12·8-s + (1.22 − 2.12i)10-s + (0.756 + 1.31i)11-s + (−2.58 − 4.48i)13-s + (3.50 + 3.38i)14-s − 4.84·16-s + (−0.774 + 1.34i)17-s + (−1.25 − 2.16i)19-s + (0.927 − 1.60i)20-s + (1.39 + 2.41i)22-s + (−3.68 + 6.37i)23-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.695·4-s + (0.298 − 0.516i)5-s + (0.719 + 0.694i)7-s − 0.396·8-s + (0.388 − 0.673i)10-s + (0.228 + 0.395i)11-s + (−0.717 − 1.24i)13-s + (0.936 + 0.904i)14-s − 1.21·16-s + (−0.187 + 0.325i)17-s + (−0.287 − 0.497i)19-s + (0.207 − 0.359i)20-s + (0.296 + 0.514i)22-s + (−0.767 + 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 + 0.0717i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.997 + 0.0717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15929 - 0.0775675i\)
\(L(\frac12)\) \(\approx\) \(2.15929 - 0.0775675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 - 1.83i)T \)
good2 \( 1 - 1.84T + 2T^{2} \)
5 \( 1 + (-0.667 + 1.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.756 - 1.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.25 + 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0309 + 0.0536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.84T + 31T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.51T + 47T^{2} \)
53 \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (2.80 - 4.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.703 + 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.57111981126569907674248360555, −12.05243608619042282350602575206, −10.91211513277231712574129966754, −9.512369323713665848353340920504, −8.558497574425318690071729101394, −7.21656457941564696111379988881, −5.57356098695714097027824682001, −5.25205267103630476225867562446, −3.88529083296901779370878286251, −2.28376037789533355383643127821, 2.38556131635350075837466356995, 3.99612609350324921865435446979, 4.78998226521223428600841376646, 6.18899723092342026367420100804, 7.01186987932375955997739123880, 8.467068523945794582047117828402, 9.775151985838219452618613454190, 10.94593661308677576117690380791, 11.76338862720031835608877717562, 12.68132403091468865552722704835

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