Properties

Label 2-189-189.5-c1-0-15
Degree $2$
Conductor $189$
Sign $0.992 + 0.125i$
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  + (1.10 − 0.195i)2-s + (1.47 + 0.909i)3-s + (−0.686 + 0.249i)4-s + (0.705 − 4.00i)5-s + (1.81 + 0.721i)6-s + (2.51 + 0.825i)7-s + (−2.66 + 1.53i)8-s + (1.34 + 2.68i)9-s − 4.57i·10-s + (−4.26 + 0.751i)11-s + (−1.23 − 0.256i)12-s + (0.646 − 0.770i)13-s + (2.95 + 0.423i)14-s + (4.68 − 5.25i)15-s + (−1.53 + 1.28i)16-s − 2.69·17-s + ⋯
L(s)  = 1  + (0.784 − 0.138i)2-s + (0.850 + 0.525i)3-s + (−0.343 + 0.124i)4-s + (0.315 − 1.78i)5-s + (0.740 + 0.294i)6-s + (0.950 + 0.311i)7-s + (−0.941 + 0.543i)8-s + (0.447 + 0.894i)9-s − 1.44i·10-s + (−1.28 + 0.226i)11-s + (−0.357 − 0.0740i)12-s + (0.179 − 0.213i)13-s + (0.788 + 0.113i)14-s + (1.20 − 1.35i)15-s + (−0.384 + 0.322i)16-s − 0.653·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97973 - 0.124346i\)
\(L(\frac12)\) \(\approx\) \(1.97973 - 0.124346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.909i)T \)
7 \( 1 + (-2.51 - 0.825i)T \)
good2 \( 1 + (-1.10 + 0.195i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (-0.705 + 4.00i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (4.26 - 0.751i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.646 + 0.770i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 5.58iT - 19T^{2} \)
23 \( 1 + (-2.40 + 2.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.209 + 0.249i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.235 - 0.646i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.72 + 2.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.76 + 4.83i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.15 + 1.14i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.47 + 0.537i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.919 + 0.530i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.43 - 7.08i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.79 + 7.67i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.201 + 1.14i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.98 - 5.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.91 - 2.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.388 - 2.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.89 + 6.62i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (3.48 - 9.56i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.84257238306680256990969241211, −11.99760618428809955789055061400, −10.49814315489604482252979664937, −9.308929920698427283983040227299, −8.450956945666835981889969641391, −8.072872505662243693748429711972, −5.36827641410270572268529293093, −4.99893476475177890322738283333, −3.96623046584830307201134373133, −2.16085327213744513506225937269, 2.45424570176506586036924306705, 3.48593253696666023190670357766, 5.00933666994119626855969672756, 6.45954137393303328717579961920, 7.24765823319291217348600715028, 8.377364710682522505224916948858, 9.656317425850881187786515779305, 10.72180510627699287996659546467, 11.60114341627521111883722663582, 13.29932872248548354105135641012

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