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Properties

Label 2-189-189.38-c1-0-16
Degree $2$
Conductor $189$
Sign $-0.879 + 0.475i$
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  + (−2.27 − 0.401i)2-s + (1.22 − 1.22i)3-s + (3.13 + 1.14i)4-s + (−0.595 − 3.37i)5-s + (−3.27 + 2.29i)6-s + (−2.64 − 0.0174i)7-s + (−2.66 − 1.54i)8-s + (0.00659 − 2.99i)9-s + 7.91i·10-s + (1.05 + 0.185i)11-s + (5.23 − 2.43i)12-s + (2.65 + 3.15i)13-s + (6.01 + 1.10i)14-s + (−4.85 − 3.41i)15-s + (0.344 + 0.288i)16-s − 6.74·17-s + ⋯
L(s)  = 1  + (−1.60 − 0.283i)2-s + (0.707 − 0.706i)3-s + (1.56 + 0.570i)4-s + (−0.266 − 1.50i)5-s + (−1.33 + 0.935i)6-s + (−0.999 − 0.00658i)7-s + (−0.943 − 0.544i)8-s + (0.00219 − 0.999i)9-s + 2.50i·10-s + (0.316 + 0.0558i)11-s + (1.51 − 0.702i)12-s + (0.735 + 0.875i)13-s + (1.60 + 0.294i)14-s + (−1.25 − 0.880i)15-s + (0.0860 + 0.0721i)16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128387 - 0.508084i\)
\(L(\frac12)\) \(\approx\) \(0.128387 - 0.508084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
7 \( 1 + (2.64 + 0.0174i)T \)
good2 \( 1 + (2.27 + 0.401i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.595 + 3.37i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-1.05 - 0.185i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-2.65 - 3.15i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 6.74T + 17T^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + (-0.748 - 0.892i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.58 - 1.88i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.16 + 8.68i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.357 - 0.618i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.44 + 4.56i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.59 + 1.30i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.83 - 0.669i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-6.35 - 3.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.57 + 3.84i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.34 - 6.43i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.361 + 2.05i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.33 + 0.773i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.58 + 5.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.698 - 3.95i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.13 + 5.14i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (0.832 + 2.28i)T + (-74.3 + 62.3i)T^{2} \)
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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

−12.02974537872902904982542534525, −11.15939616833397433393579232026, −9.534564741978647619169963141037, −9.020075854102024910165904208581, −8.589168087738302233397300303180, −7.35052805977405836375729333418, −6.40143734806208567110851523014, −4.15118981631291045863367296289, −2.21530947919689597102336797932, −0.71321437079104512652232924076, 2.58062891293300325642476724767, 3.74603884911007646042825121788, 6.25701844729181351186452976540, 7.05480426099509976644824560230, 8.146590808944854505869160402858, 9.007429473607375937932560369180, 10.01647575886671000386464693754, 10.57580039355575601536337695952, 11.26379614117178193367945204109, 13.11679527004660570521848803173

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