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Properties

Label 2-189-189.68-c1-0-13
Degree $2$
Conductor $189$
Sign $0.999 - 0.00828i$
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  + (−0.218 + 0.601i)2-s + (0.0957 − 1.72i)3-s + (1.21 + 1.02i)4-s + (0.158 − 0.0577i)5-s + (1.01 + 0.436i)6-s + (2.63 + 0.280i)7-s + (−1.98 + 1.14i)8-s + (−2.98 − 0.331i)9-s + 0.108i·10-s + (0.924 − 2.53i)11-s + (1.88 − 2.00i)12-s + (1.29 + 0.228i)13-s + (−0.744 + 1.52i)14-s + (−0.0846 − 0.279i)15-s + (0.297 + 1.68i)16-s + 3.38·17-s + ⋯
L(s)  = 1  + (−0.154 + 0.425i)2-s + (0.0552 − 0.998i)3-s + (0.609 + 0.511i)4-s + (0.0709 − 0.0258i)5-s + (0.416 + 0.178i)6-s + (0.994 + 0.106i)7-s + (−0.703 + 0.406i)8-s + (−0.993 − 0.110i)9-s + 0.0341i·10-s + (0.278 − 0.765i)11-s + (0.544 − 0.579i)12-s + (0.359 + 0.0633i)13-s + (−0.199 + 0.406i)14-s + (−0.0218 − 0.0722i)15-s + (0.0742 + 0.421i)16-s + 0.821·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.999 - 0.00828i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.999 - 0.00828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33111 + 0.00551299i\)
\(L(\frac12)\) \(\approx\) \(1.33111 + 0.00551299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0957 + 1.72i)T \)
7 \( 1 + (-2.63 - 0.280i)T \)
good2 \( 1 + (0.218 - 0.601i)T + (-1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.158 + 0.0577i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.924 + 2.53i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.29 - 0.228i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 - 3.38T + 17T^{2} \)
19 \( 1 + 1.40iT - 19T^{2} \)
23 \( 1 + (1.02 + 0.181i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (8.68 - 1.53i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.90 + 3.46i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.11 + 3.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.238 - 1.35i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (7.25 - 6.09i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (5.47 - 4.59i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.88 - 1.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.918 + 5.20i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.98 + 8.32i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.357 - 0.130i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.64 - 3.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.0 + 6.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.3 - 4.11i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.97 - 16.8i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-2.45 - 2.92i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.51274234363148326359356986229, −11.49473599519151016671364129963, −11.15626386592163844620160838543, −9.188434812765535692459926141848, −8.112993052348100267170286554837, −7.65003741097396382948331989334, −6.41949057944053436248697801419, −5.51425413744846168686805553094, −3.36839199029198017905056509677, −1.78628036430945268468162941275, 1.89769466377057157758367401005, 3.60424250065479937655676755980, 4.98450044656719050077544537439, 6.04235001899373421050267470800, 7.56546781486171590872782007243, 8.795205469943935940750622650111, 9.965277014747345108320107888263, 10.42582645816907485049605304561, 11.54142709928718501267973460471, 12.02418336843954455249746746795

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