Properties

Label 2-189-189.5-c1-0-18
Degree $2$
Conductor $189$
Sign $0.710 + 0.704i$
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  + (1.77 − 0.312i)2-s + (0.626 − 1.61i)3-s + (1.17 − 0.426i)4-s + (−0.155 + 0.884i)5-s + (0.606 − 3.06i)6-s + (2.64 − 0.129i)7-s + (−1.17 + 0.678i)8-s + (−2.21 − 2.02i)9-s + 1.61i·10-s + (−3.83 + 0.675i)11-s + (0.0457 − 2.15i)12-s + (−1.72 + 2.05i)13-s + (4.64 − 1.05i)14-s + (1.33 + 0.806i)15-s + (−3.78 + 3.17i)16-s + 7.46·17-s + ⋯
L(s)  = 1  + (1.25 − 0.221i)2-s + (0.361 − 0.932i)3-s + (0.585 − 0.213i)4-s + (−0.0697 + 0.395i)5-s + (0.247 − 1.24i)6-s + (0.998 − 0.0490i)7-s + (−0.415 + 0.240i)8-s + (−0.738 − 0.674i)9-s + 0.511i·10-s + (−1.15 + 0.203i)11-s + (0.0132 − 0.623i)12-s + (−0.479 + 0.570i)13-s + (1.24 − 0.282i)14-s + (0.343 + 0.208i)15-s + (−0.945 + 0.793i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.710 + 0.704i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.710 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04055 - 0.840252i\)
\(L(\frac12)\) \(\approx\) \(2.04055 - 0.840252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.626 + 1.61i)T \)
7 \( 1 + (-2.64 + 0.129i)T \)
good2 \( 1 + (-1.77 + 0.312i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.155 - 0.884i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (3.83 - 0.675i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.72 - 2.05i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
19 \( 1 + 4.84iT - 19T^{2} \)
23 \( 1 + (5.01 - 5.97i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.24 + 3.87i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.0550 - 0.151i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.45 + 4.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.78 - 2.33i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.36 - 1.95i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (3.75 + 1.36i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.73 - 1.58i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.24 + 3.56i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.92 + 8.02i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.25 + 7.12i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.65 + 2.68i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.96 - 4.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.02 + 11.4i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.47 + 6.27i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 7.39T + 89T^{2} \)
97 \( 1 + (0.434 - 1.19i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.55594092573281885975619704090, −11.80014553539198748402579802996, −11.00393161238656306448103029993, −9.425942130301355023422155710386, −7.999977793963266726514442554483, −7.34346346509545862573465761304, −5.84859582261901273325524360937, −4.92077271690687664582357039435, −3.36197413527158575364730030049, −2.16007842320567314331432626567, 2.84404473051004809288525240983, 4.10930031819252046260613436256, 5.17667610571769117668033180906, 5.63694920887184019921712268608, 7.74698479452509243845428931083, 8.473967053188466530910014220567, 9.928485950979813830549505959410, 10.69067229227611521555424396174, 12.08329066310645287157127021808, 12.68223807070413716381864392554

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