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Properties

Label 2-189-189.101-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.449 + 0.893i$
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  + (−1.22 + 1.45i)2-s + (−1.12 + 1.32i)3-s + (−0.280 − 1.59i)4-s + (−0.691 + 0.580i)5-s + (−0.555 − 3.24i)6-s + (−0.448 + 2.60i)7-s + (−0.629 − 0.363i)8-s + (−0.489 − 2.95i)9-s − 1.71i·10-s + (−2.24 + 2.67i)11-s + (2.41 + 1.41i)12-s + (1.35 − 3.72i)13-s + (−3.25 − 3.84i)14-s + (0.00822 − 1.56i)15-s + (4.33 − 1.57i)16-s − 0.467·17-s + ⋯
L(s)  = 1  + (−0.864 + 1.03i)2-s + (−0.646 + 0.762i)3-s + (−0.140 − 0.796i)4-s + (−0.309 + 0.259i)5-s + (−0.226 − 1.32i)6-s + (−0.169 + 0.985i)7-s + (−0.222 − 0.128i)8-s + (−0.163 − 0.986i)9-s − 0.542i·10-s + (−0.677 + 0.807i)11-s + (0.698 + 0.408i)12-s + (0.375 − 1.03i)13-s + (−0.868 − 1.02i)14-s + (0.00212 − 0.403i)15-s + (1.08 − 0.394i)16-s − 0.113·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.449 + 0.893i$
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: \((2,\ 189,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148779 - 0.241543i\)
\(L(\frac12)\) \(\approx\) \(0.148779 - 0.241543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 - 1.32i)T \)
7 \( 1 + (0.448 - 2.60i)T \)
good2 \( 1 + (1.22 - 1.45i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.691 - 0.580i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.24 - 2.67i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.35 + 3.72i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + (1.19 - 3.28i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.79 + 4.92i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (6.86 - 1.20i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.92 - 8.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.71 + 0.987i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.54 - 8.76i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.579 - 3.28i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-8.75 - 5.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.04 + 1.10i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.0601 - 0.0106i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.85 + 6.58i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.19 - 1.26i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.24 - 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.23 - 6.07i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.6 + 3.86i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 3.03T + 89T^{2} \)
97 \( 1 + (5.02 + 0.886i)T + (91.1 + 33.1i)T^{2} \)
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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

−13.08154918000914081542825552503, −12.10361795132910769095296263164, −11.08584262978744492928102428889, −9.962207477695216339448996476009, −9.250651819945500861600366133347, −8.190586817300586998577323988342, −7.12987648094561566049493610286, −5.96655751997408982153244876612, −5.12874234844436603278331610803, −3.25187447128745461226650880438, 0.35069715895337683308750955795, 1.90977762944796059933336640531, 3.77859720742792644788932965067, 5.56534571575598459288060326378, 6.86505682171820176589693222050, 8.023730931251523748458182059150, 8.905391198136339884525595401208, 10.38583502357497265484393601373, 10.80331961197661633080392860617, 11.74981234752454922641978848206

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