Properties

Label 2-189-189.4-c1-0-4
Degree $2$
Conductor $189$
Sign $0.139 - 0.990i$
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  + (0.340 + 1.93i)2-s + (−0.913 − 1.47i)3-s + (−1.74 + 0.633i)4-s + (1.35 − 0.494i)5-s + (2.53 − 2.26i)6-s + (−0.147 + 2.64i)7-s + (0.143 + 0.249i)8-s + (−1.33 + 2.68i)9-s + (1.41 + 2.45i)10-s + (5.77 + 2.10i)11-s + (2.52 + 1.98i)12-s + (1.44 − 0.527i)13-s + (−5.15 + 0.616i)14-s + (−1.96 − 1.54i)15-s + (−3.27 + 2.74i)16-s + (−3.07 − 5.32i)17-s + ⋯
L(s)  = 1  + (0.241 + 1.36i)2-s + (−0.527 − 0.849i)3-s + (−0.870 + 0.316i)4-s + (0.607 − 0.221i)5-s + (1.03 − 0.925i)6-s + (−0.0555 + 0.998i)7-s + (0.0508 + 0.0881i)8-s + (−0.444 + 0.895i)9-s + (0.449 + 0.777i)10-s + (1.74 + 0.633i)11-s + (0.728 + 0.572i)12-s + (0.401 − 0.146i)13-s + (−1.37 + 0.164i)14-s + (−0.508 − 0.399i)15-s + (−0.818 + 0.686i)16-s + (−0.745 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973400 + 0.845747i\)
\(L(\frac12)\) \(\approx\) \(0.973400 + 0.845747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.913 + 1.47i)T \)
7 \( 1 + (0.147 - 2.64i)T \)
good2 \( 1 + (-0.340 - 1.93i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.35 + 0.494i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-5.77 - 2.10i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.44 + 0.527i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.07 + 5.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.261 + 0.452i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.43 + 8.14i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.17 + 1.51i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.80 - 1.74i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 - 0.210T + 37T^{2} \)
41 \( 1 + (3.94 - 1.43i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.0927 + 0.526i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-9.52 - 3.46i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-2.29 + 3.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.47 - 2.91i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.25 + 1.91i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.91 + 10.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.605 + 1.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.51T + 73T^{2} \)
79 \( 1 + (-0.432 - 2.45i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.29 + 0.836i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.13 + 3.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.30 + 13.0i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.96238956693340771182419480498, −12.00419568625169286268163386220, −11.13362657425731361326210599478, −9.299873421278453475493697843328, −8.643104260335503692906796161248, −7.23055355861346620128140589124, −6.51523098399371130326994543574, −5.74843746743365524853836902700, −4.70964745486795166559622307548, −2.04001930067759831939287519146, 1.45937369002613672963689530131, 3.68871707147998142956330167853, 4.00953140042159815862202424437, 5.80653721791515215469656029353, 6.87968768887456194786039806132, 8.985462307426586317865229254586, 9.709039010170859017152185144853, 10.66206219696890938536412909249, 11.19863896416151756191657510013, 11.97892051008992689984002230914

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