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Properties

Label 2-189-189.184-c1-0-12
Degree $2$
Conductor $189$
Sign $0.515 + 0.856i$
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  + (−0.202 − 1.14i)2-s + (1.53 − 0.810i)3-s + (0.599 − 0.218i)4-s + (−0.490 + 2.78i)5-s + (−1.24 − 1.59i)6-s + (2.49 + 0.887i)7-s + (−1.53 − 2.66i)8-s + (1.68 − 2.48i)9-s + 3.29·10-s + (0.740 + 4.20i)11-s + (0.740 − 0.819i)12-s + (−4.36 − 3.65i)13-s + (0.514 − 3.04i)14-s + (1.50 + 4.65i)15-s + (−1.77 + 1.49i)16-s − 5.92·17-s + ⋯
L(s)  = 1  + (−0.143 − 0.812i)2-s + (0.883 − 0.467i)3-s + (0.299 − 0.109i)4-s + (−0.219 + 1.24i)5-s + (−0.506 − 0.651i)6-s + (0.942 + 0.335i)7-s + (−0.544 − 0.942i)8-s + (0.562 − 0.826i)9-s + 1.04·10-s + (0.223 + 1.26i)11-s + (0.213 − 0.236i)12-s + (−1.20 − 1.01i)13-s + (0.137 − 0.813i)14-s + (0.388 + 1.20i)15-s + (−0.444 + 0.372i)16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35457 - 0.766022i\)
\(L(\frac12)\) \(\approx\) \(1.35457 - 0.766022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.810i)T \)
7 \( 1 + (-2.49 - 0.887i)T \)
good2 \( 1 + (0.202 + 1.14i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.490 - 2.78i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.740 - 4.20i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (4.36 + 3.65i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 5.92T + 17T^{2} \)
19 \( 1 + 4.28T + 19T^{2} \)
23 \( 1 + (0.619 + 0.519i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.10 + 2.60i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.81 + 0.659i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-4.35 - 7.53i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.202 + 0.169i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.02 + 0.374i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.38 + 0.867i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.32 + 4.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.70 - 5.62i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-7.18 - 2.61i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.01 - 5.72i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.32 + 10.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.61 - 7.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.314 - 1.78i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.15 + 3.48i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 0.991T + 89T^{2} \)
97 \( 1 + (1.72 + 0.628i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.25661764886336769367340122794, −11.46321312236413929240238528780, −10.42455552505245378973608168950, −9.732581725148352213934065800171, −8.329734991361592348437622862050, −7.24119569257009884289748645025, −6.55763184915740507027242048925, −4.38212932604694879180313002780, −2.72761917713893456870446960432, −2.12365992100727019636365843901, 2.20015287705011660817250237849, 4.22722489810950838828128793528, 5.06906364078607968871212815565, 6.71782515918488999189892460432, 7.912468504936600620853640213431, 8.597781839115367401228630846763, 9.158970106433261693357579488636, 10.84065286413587688798632421330, 11.66923536184302277066637562675, 12.88385437674803847845303626882

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