Properties

Label 2-189-27.22-c1-0-16
Degree $2$
Conductor $189$
Sign $0.357 + 0.934i$
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  + (2.00 − 1.68i)2-s + (1.58 + 0.708i)3-s + (0.838 − 4.75i)4-s + (−3.29 + 1.20i)5-s + (4.35 − 1.23i)6-s + (0.173 + 0.984i)7-s + (−3.69 − 6.40i)8-s + (1.99 + 2.23i)9-s + (−4.58 + 7.94i)10-s + (−0.308 − 0.112i)11-s + (4.69 − 6.92i)12-s + (−1.82 − 1.52i)13-s + (2.00 + 1.68i)14-s + (−6.06 − 0.438i)15-s + (−9.09 − 3.30i)16-s + (−1.28 + 2.22i)17-s + ⋯
L(s)  = 1  + (1.41 − 1.18i)2-s + (0.912 + 0.408i)3-s + (0.419 − 2.37i)4-s + (−1.47 + 0.536i)5-s + (1.77 − 0.505i)6-s + (0.0656 + 0.372i)7-s + (−1.30 − 2.26i)8-s + (0.665 + 0.746i)9-s + (−1.45 + 2.51i)10-s + (−0.0931 − 0.0339i)11-s + (1.35 − 1.99i)12-s + (−0.504 − 0.423i)13-s + (0.535 + 0.448i)14-s + (−1.56 − 0.113i)15-s + (−2.27 − 0.827i)16-s + (−0.310 + 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.357 + 0.934i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.357 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01956 - 1.39005i\)
\(L(\frac12)\) \(\approx\) \(2.01956 - 1.39005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.708i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (-2.00 + 1.68i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (3.29 - 1.20i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (0.308 + 0.112i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.82 + 1.52i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.28 - 2.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.50 - 4.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.785 + 4.45i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.49 - 6.29i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.70 + 9.69i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.70 + 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.18 + 4.34i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.843 - 0.306i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.875 + 4.96i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 + (-6.88 + 2.50i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.157 - 0.894i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.46 - 3.74i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (5.46 - 9.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.95 - 5.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.653 - 0.548i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.71 - 1.43i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.107 - 0.186i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.73 + 0.994i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.42834726366459194162375409153, −11.51904259037786240538384612326, −10.73167642338523269013742801421, −9.851346046362981416225742763470, −8.382726662975516954703373760617, −7.20887978032191382487354855960, −5.45170152043674487232844945261, −4.15046619133655601980074859764, −3.52309701316089119627530178906, −2.37170587631145627926397375762, 3.12850000187892810468746484750, 4.13398292211699553288834786801, 5.00240366930489194982405359500, 6.83573129124422614549739778192, 7.44794392711593229406724516124, 8.143027727845031757768564834698, 9.234383616766605965606959531768, 11.53375125790143722124077710731, 12.09138188893855082310792036096, 13.14596673058948215974128513132

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