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Properties

Label 2-189-189.88-c1-0-8
Degree $2$
Conductor $189$
Sign $0.878 - 0.478i$
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  + (0.421 + 0.353i)2-s + (0.959 + 1.44i)3-s + (−0.294 − 1.67i)4-s + (0.790 − 0.663i)5-s + (−0.105 + 0.946i)6-s + (1.85 + 1.88i)7-s + (1.01 − 1.76i)8-s + (−1.15 + 2.76i)9-s + 0.567·10-s + (−1.81 − 1.52i)11-s + (2.12 − 2.02i)12-s + (1.20 + 0.439i)13-s + (0.117 + 1.44i)14-s + (1.71 + 0.503i)15-s + (−2.14 + 0.779i)16-s − 1.63·17-s + ⋯
L(s)  = 1  + (0.297 + 0.249i)2-s + (0.553 + 0.832i)3-s + (−0.147 − 0.835i)4-s + (0.353 − 0.296i)5-s + (−0.0430 + 0.386i)6-s + (0.702 + 0.711i)7-s + (0.359 − 0.622i)8-s + (−0.386 + 0.922i)9-s + 0.179·10-s + (−0.547 − 0.459i)11-s + (0.614 − 0.585i)12-s + (0.335 + 0.121i)13-s + (0.0314 + 0.387i)14-s + (0.442 + 0.129i)15-s + (−0.535 + 0.194i)16-s − 0.395·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.878 - 0.478i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59229 + 0.405637i\)
\(L(\frac12)\) \(\approx\) \(1.59229 + 0.405637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.959 - 1.44i)T \)
7 \( 1 + (-1.85 - 1.88i)T \)
good2 \( 1 + (-0.421 - 0.353i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.790 + 0.663i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (1.81 + 1.52i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.20 - 0.439i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
23 \( 1 + (3.76 + 1.36i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (5.04 - 1.83i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.00 + 5.68i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-5.33 + 9.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.546 - 0.198i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.125 + 0.712i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.16 + 12.2i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (5.06 - 8.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.04 + 2.56i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.978 - 5.54i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.92 - 2.45i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.93 - 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.83 - 8.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.12 - 5.98i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.769 + 0.280i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (-1.46 + 8.28i)T + (-91.1 - 33.1i)T^{2} \)
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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

−13.00757467015923607586265074026, −11.37822949474501517852299819074, −10.65289191088529323445475773621, −9.509855576482682084891066198722, −8.888226011057868378478231086434, −7.70443756843495878272057464124, −5.83946080691741089321890422659, −5.26223839476458306582129407107, −4.04770986857824179815184716403, −2.13511158208489780362994147728, 1.99102202518102968018372420612, 3.33394068549600057490353974632, 4.67960550917026091093434981447, 6.40973097913502929127070085570, 7.66492021185290610158891113593, 8.056473313867095670524942199806, 9.389496044426061887277028333725, 10.73707277358988025308009861025, 11.72422638649756322200312640395, 12.62822135863060517384650682630

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