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Properties

Label 2-189-63.4-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.876 + 0.482i$
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  + (−0.920 + 1.59i)2-s + (−0.695 − 1.20i)4-s − 1.33·5-s + (−2.54 + 0.728i)7-s − 1.12·8-s + (1.22 − 2.12i)10-s − 1.51·11-s + (−2.58 + 4.48i)13-s + (1.17 − 4.72i)14-s + (2.42 − 4.19i)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 + 7.36·23-s + ⋯
L(s)  = 1  + (−0.650 + 1.12i)2-s + (−0.347 − 0.601i)4-s − 0.596·5-s + (−0.961 + 0.275i)7-s − 0.396·8-s + (0.388 − 0.673i)10-s − 0.456·11-s + (−0.717 + 1.24i)13-s + (0.315 − 1.26i)14-s + (0.605 − 1.04i)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 + 1.53·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.876 + 0.482i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.876 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0853641 - 0.332237i\)
\(L(\frac12)\) \(\approx\) \(0.0853641 - 0.332237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.728i)T \)
good2 \( 1 + (0.920 - 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
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 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (4.75 - 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.22 + 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
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

−13.10048818085696366466085675311, −12.20472897024096506141497040155, −11.16749713087238543228671532732, −9.690009091391917467720593876787, −9.055587215689502072818242161725, −7.968868212779612737500920778008, −6.98486613380749382289544685881, −6.27586416674880033885751495143, −4.75273592288863850352114444931, −3.00043713575327089756049846821, 0.35617371768237503392341351278, 2.63473575069796585917130883873, 3.68395300331636036103125010239, 5.50025440818610696187411449260, 7.01930763146187339845100314192, 8.161244934194465721823351529513, 9.299386362409250385406381334548, 10.20022254632082446477539669747, 10.80363640161905766818006825701, 11.96401365459806877574374087953

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