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Properties

Label 2-189-189.101-c1-0-11
Degree $2$
Conductor $189$
Sign $0.998 - 0.0554i$
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  + (−1.07 + 1.28i)2-s + (−1.69 + 0.345i)3-s + (−0.142 − 0.809i)4-s + (2.87 − 2.40i)5-s + (1.38 − 2.55i)6-s + (−1.86 − 1.87i)7-s + (−1.71 − 0.989i)8-s + (2.76 − 1.17i)9-s + 6.29i·10-s + (3.07 − 3.66i)11-s + (0.521 + 1.32i)12-s + (−1.10 + 3.04i)13-s + (4.42 − 0.369i)14-s + (−4.04 + 5.07i)15-s + (4.66 − 1.69i)16-s + 3.86·17-s + ⋯
L(s)  = 1  + (−0.763 + 0.909i)2-s + (−0.979 + 0.199i)3-s + (−0.0713 − 0.404i)4-s + (1.28 − 1.07i)5-s + (0.566 − 1.04i)6-s + (−0.704 − 0.709i)7-s + (−0.606 − 0.349i)8-s + (0.920 − 0.390i)9-s + 1.99i·10-s + (0.926 − 1.10i)11-s + (0.150 + 0.382i)12-s + (−0.307 + 0.844i)13-s + (1.18 − 0.0987i)14-s + (−1.04 + 1.31i)15-s + (1.16 − 0.424i)16-s + 0.938·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.998 - 0.0554i$
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: \((2,\ 189,\ (\ :1/2),\ 0.998 - 0.0554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672109 + 0.0186504i\)
\(L(\frac12)\) \(\approx\) \(0.672109 + 0.0186504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.345i)T \)
7 \( 1 + (1.86 + 1.87i)T \)
good2 \( 1 + (1.07 - 1.28i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-2.87 + 2.40i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-3.07 + 3.66i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.10 - 3.04i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.86T + 17T^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (-0.124 + 0.342i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.29 + 6.30i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.692 - 0.122i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.75 + 4.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.793 - 0.288i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.0136 + 0.0775i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.617 + 3.50i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-7.25 - 4.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 0.495i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.82 - 0.497i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.09 + 2.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (14.1 - 8.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.4 - 6.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 0.844i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.7 - 4.64i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 8.72T + 89T^{2} \)
97 \( 1 + (4.18 + 0.738i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.57610105345792720397501884011, −11.64918221112772963174981091846, −10.11224208017015743732417275669, −9.544020631147351552507916554338, −8.758721744942771030569993395187, −7.25237907169834183298055061416, −6.17912718979009943835821644864, −5.69975350584101068016624964667, −4.01287601133145057086944115556, −0.948614694647262381286534271821, 1.70255150418066574753903759887, 2.96964218192297453382634073805, 5.42877665229551633522706235397, 6.21862056252916745860099377696, 7.20097994089055800652751448419, 9.168740495156918664914926110158, 9.988601987206820757830785059066, 10.30768401128214469662444082703, 11.44879266067804835054657015232, 12.29726101656925470465184487132

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