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Properties

Label 2-189-189.101-c1-0-14
Degree $2$
Conductor $189$
Sign $0.553 + 0.832i$
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  + (0.905 − 1.07i)2-s + (−1.67 + 0.438i)3-s + (0.00302 + 0.0171i)4-s + (1.61 − 1.35i)5-s + (−1.04 + 2.20i)6-s + (1.08 − 2.41i)7-s + (2.46 + 1.42i)8-s + (2.61 − 1.46i)9-s − 2.97i·10-s + (−2.38 + 2.83i)11-s + (−0.0125 − 0.0274i)12-s + (2.27 − 6.25i)13-s + (−1.61 − 3.35i)14-s + (−2.11 + 2.98i)15-s + (3.72 − 1.35i)16-s − 3.23·17-s + ⋯
L(s)  = 1  + (0.639 − 0.762i)2-s + (−0.967 + 0.252i)3-s + (0.00151 + 0.00857i)4-s + (0.722 − 0.606i)5-s + (−0.426 + 0.899i)6-s + (0.411 − 0.911i)7-s + (0.869 + 0.502i)8-s + (0.872 − 0.489i)9-s − 0.939i·10-s + (−0.717 + 0.855i)11-s + (−0.00363 − 0.00791i)12-s + (0.631 − 1.73i)13-s + (−0.432 − 0.897i)14-s + (−0.545 + 0.769i)15-s + (0.931 − 0.339i)16-s − 0.785·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27114 - 0.681501i\)
\(L(\frac12)\) \(\approx\) \(1.27114 - 0.681501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.438i)T \)
7 \( 1 + (-1.08 + 2.41i)T \)
good2 \( 1 + (-0.905 + 1.07i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.61 + 1.35i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.38 - 2.83i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.27 + 6.25i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
19 \( 1 - 5.32iT - 19T^{2} \)
23 \( 1 + (0.239 - 0.658i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.50 - 4.12i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.47 - 0.260i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.24 - 5.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.4 + 3.81i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.29 + 7.32i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.04 - 5.90i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.766 + 0.442i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.85 - 2.13i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.32 - 1.29i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.05 + 6.76i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.83 - 2.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.65 - 1.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.29 + 6.11i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (11.4 - 4.16i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 + (-6.85 - 1.20i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.63684467292398391281358127580, −11.47385621609726229431370911852, −10.36238706312607206978322140157, −10.24661650489078254812611356611, −8.319296773898199773496627513664, −7.18093695103602881238986380372, −5.53141611168732913935059808168, −4.86437722792925417172326804595, −3.62597191515603494371912329280, −1.56688288485419397182751855067, 2.07281348564212741926284495801, 4.51804723031525360790204992839, 5.52650723546416200854126764369, 6.34738257574250825082803790445, 6.94069340072136108944155829227, 8.569835736163330821745037938505, 9.915307889757419682314392112261, 11.06407458476989420863372537033, 11.53865742374886501425044168318, 13.07972810252205557780179742760

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