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Properties

Label 2-189-7.4-c1-0-4
Degree $2$
Conductor $189$
Sign $0.757 - 0.653i$
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  + (0.380 + 0.658i)2-s + (0.710 − 1.23i)4-s + (1.59 + 2.75i)5-s + (−2.56 + 0.658i)7-s + 2.60·8-s + (−1.21 + 2.09i)10-s + (1.11 − 1.93i)11-s + 3.70·13-s + (−1.40 − 1.43i)14-s + (−0.430 − 0.746i)16-s + (−2.80 + 4.85i)17-s + (−2.21 − 3.82i)19-s + 4.52·20-s + 1.70·22-s + (−0.471 − 0.816i)23-s + ⋯
L(s)  = 1  + (0.269 + 0.465i)2-s + (0.355 − 0.615i)4-s + (0.711 + 1.23i)5-s + (−0.968 + 0.249i)7-s + 0.920·8-s + (−0.382 + 0.663i)10-s + (0.337 − 0.584i)11-s + 1.02·13-s + (−0.376 − 0.384i)14-s + (−0.107 − 0.186i)16-s + (−0.679 + 1.17i)17-s + (−0.507 − 0.878i)19-s + 1.01·20-s + 0.363·22-s + (−0.0982 − 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.757 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43566 + 0.533783i\)
\(L(\frac12)\) \(\approx\) \(1.43566 + 0.533783i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.56 - 0.658i)T \)
good2 \( 1 + (-0.380 - 0.658i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.59 - 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.11 + 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.70T + 13T^{2} \)
17 \( 1 + (2.80 - 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 3.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.471 + 0.816i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + (-2.85 + 4.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.56 + 2.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.98T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 + (0.112 + 0.195i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.33 + 9.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.02 - 1.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.92 - 5.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.71 - 6.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.26T + 71T^{2} \)
73 \( 1 + (3.77 - 6.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.41 - 5.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.11T + 83T^{2} \)
89 \( 1 + (-4.86 - 8.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.842T + 97T^{2} \)
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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

−13.21127730334819209769950652315, −11.36102449775286229676069431917, −10.72686523309820910900740614405, −9.894964375118158660220916264576, −8.728203315906913205183898444683, −7.02424117395271113149223290533, −6.30855650846792055215385772322, −5.78888646380277477436126506371, −3.75142663198476890107936911459, −2.21574931045802993404890900932, 1.75562215354869559880831019856, 3.48145875187955992746860293345, 4.66878324491480102281295224713, 6.11202225945612341545576712914, 7.25794101352671652131950199405, 8.641703496719825040839786283806, 9.452722215915119019435383632608, 10.52050271071881375961147794072, 11.74523700019565292406525596386, 12.55835834086771301774830027719

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