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Properties

Label 2-189-189.101-c1-0-9
Degree $2$
Conductor $189$
Sign $0.690 - 0.722i$
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  + (0.204 − 0.244i)2-s + (1.66 + 0.471i)3-s + (0.329 + 1.86i)4-s + (−2.18 + 1.83i)5-s + (0.456 − 0.310i)6-s + (0.468 − 2.60i)7-s + (1.07 + 0.621i)8-s + (2.55 + 1.57i)9-s + 0.911i·10-s + (−2.19 + 2.61i)11-s + (−0.333 + 3.27i)12-s + (1.17 − 3.21i)13-s + (−0.539 − 0.647i)14-s + (−4.51 + 2.02i)15-s + (−3.19 + 1.16i)16-s + 4.66·17-s + ⋯
L(s)  = 1  + (0.144 − 0.172i)2-s + (0.962 + 0.272i)3-s + (0.164 + 0.934i)4-s + (−0.979 + 0.821i)5-s + (0.186 − 0.126i)6-s + (0.176 − 0.984i)7-s + (0.380 + 0.219i)8-s + (0.851 + 0.524i)9-s + 0.288i·10-s + (−0.660 + 0.787i)11-s + (−0.0961 + 0.944i)12-s + (0.324 − 0.891i)13-s + (−0.144 − 0.173i)14-s + (−1.16 + 0.523i)15-s + (−0.798 + 0.290i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.690 - 0.722i$
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: \((2,\ 189,\ (\ :1/2),\ 0.690 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41658 + 0.605709i\)
\(L(\frac12)\) \(\approx\) \(1.41658 + 0.605709i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.471i)T \)
7 \( 1 + (-0.468 + 2.60i)T \)
good2 \( 1 + (-0.204 + 0.244i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (2.18 - 1.83i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.19 - 2.61i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.17 + 3.21i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 4.66T + 17T^{2} \)
19 \( 1 + 5.52iT - 19T^{2} \)
23 \( 1 + (-0.470 + 1.29i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.81 + 7.73i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.17 + 0.384i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.882 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.96 + 1.80i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.57 - 8.95i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.532 + 3.01i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.30 + 0.756i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.44 + 2.34i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-9.69 - 1.70i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.34 - 6.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (9.64 - 5.56i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.11 - 2.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.75 - 7.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.72 - 2.44i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-2.47 - 0.436i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.91134242417539123724554748911, −11.64464000929094886018040688539, −10.75031186361926560518566795483, −9.899277214430616907807765695669, −8.265973971873224823894138269231, −7.64618791023704173711181507984, −7.09666729657710794602825953955, −4.59844371438084166140398797684, −3.61946272831048426171300728058, −2.73904146822566880057601413505, 1.57120247218581575094770730900, 3.44797500262462352451651922664, 4.91691935690985509614421413307, 6.04124762268617580942468689748, 7.53459861656874431027279024378, 8.442308742388670077553436326524, 9.165506461079790192236853476856, 10.37487010332755060186129363301, 11.71794576680270836539577130334, 12.40834818559789547096021531127

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