Properties

Label 2-189-189.101-c1-0-6
Degree $2$
Conductor $189$
Sign $-0.402 - 0.915i$
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  + (−0.575 + 0.686i)2-s + (1.25 + 1.18i)3-s + (0.207 + 1.17i)4-s + (0.100 − 0.0844i)5-s + (−1.54 + 0.178i)6-s + (−1.70 + 2.02i)7-s + (−2.48 − 1.43i)8-s + (0.169 + 2.99i)9-s + 0.117i·10-s + (2.57 − 3.06i)11-s + (−1.14 + 1.73i)12-s + (1.02 − 2.81i)13-s + (−0.409 − 2.33i)14-s + (0.227 + 0.0134i)15-s + (0.160 − 0.0583i)16-s − 0.344·17-s + ⋯
L(s)  = 1  + (−0.407 + 0.485i)2-s + (0.726 + 0.686i)3-s + (0.103 + 0.589i)4-s + (0.0450 − 0.0377i)5-s + (−0.629 + 0.0730i)6-s + (−0.643 + 0.765i)7-s + (−0.877 − 0.506i)8-s + (0.0566 + 0.998i)9-s + 0.0372i·10-s + (0.776 − 0.925i)11-s + (−0.329 + 0.500i)12-s + (0.284 − 0.781i)13-s + (−0.109 − 0.623i)14-s + (0.0586 + 0.00346i)15-s + (0.0401 − 0.0145i)16-s − 0.0835·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.402 - 0.915i$
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: \((2,\ 189,\ (\ :1/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630715 + 0.966676i\)
\(L(\frac12)\) \(\approx\) \(0.630715 + 0.966676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 - 1.18i)T \)
7 \( 1 + (1.70 - 2.02i)T \)
good2 \( 1 + (0.575 - 0.686i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.100 + 0.0844i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.57 + 3.06i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.02 + 2.81i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.344T + 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + (-2.18 + 5.98i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.29 - 6.31i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-8.59 + 1.51i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.64 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.04 + 3.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.350 + 1.98i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.771 + 4.37i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.49 - 3.17i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.167 - 0.0608i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (4.60 + 0.811i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.83 + 7.40i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.373 + 0.215i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.31 + 0.761i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.81 - 1.52i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.81 - 2.48i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + (12.2 + 2.15i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.87838144350133566120987326875, −12.00503526314503537260966119774, −10.70684621512104008296274313277, −9.575473594610449465676754623923, −8.684110912560382498824629840213, −8.228396275420686036433545818806, −6.76891813996214211143850746462, −5.61942651120282467337216157097, −3.76740496263258838358437162130, −2.89025743223912922709913531412, 1.25332310184182280441666623128, 2.73435395099589523006524410899, 4.34484886609598857639151719799, 6.41291175955987265072158634096, 6.92327675056423041108427965450, 8.391130209093511805745055050330, 9.561911050458952332325704892092, 9.890832164407352185866222030309, 11.39587471896365330440212520118, 12.08498416413137732434642721855

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