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Properties

Label 2-189-63.11-c2-0-3
Degree $2$
Conductor $189$
Sign $-0.846 - 0.531i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.09i·2-s − 5.56·4-s + (5.67 + 3.27i)5-s + (6.63 + 2.22i)7-s − 4.83i·8-s + (−10.1 + 17.5i)10-s + (6.01 − 3.47i)11-s + (4.39 + 7.61i)13-s + (−6.86 + 20.5i)14-s − 7.29·16-s + (−28.2 − 16.3i)17-s + (−1.93 − 3.35i)19-s + (−31.5 − 18.2i)20-s + (10.7 + 18.5i)22-s + (8.88 + 5.12i)23-s + ⋯
L(s)  = 1  + 1.54i·2-s − 1.39·4-s + (1.13 + 0.655i)5-s + (0.948 + 0.317i)7-s − 0.604i·8-s + (−1.01 + 1.75i)10-s + (0.546 − 0.315i)11-s + (0.338 + 0.585i)13-s + (−0.490 + 1.46i)14-s − 0.456·16-s + (−1.66 − 0.959i)17-s + (−0.101 − 0.176i)19-s + (−1.57 − 0.912i)20-s + (0.487 + 0.844i)22-s + (0.386 + 0.222i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.846 - 0.531i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.846 - 0.531i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.507676 + 1.76318i\)
\(L(\frac12)\) \(\approx\) \(0.507676 + 1.76318i\)
\(L(2)\) 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 \( 1 + (-6.63 - 2.22i)T \)
good2 \( 1 - 3.09iT - 4T^{2} \)
5 \( 1 + (-5.67 - 3.27i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.01 + 3.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.39 - 7.61i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (28.2 + 16.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.93 + 3.35i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.88 - 5.12i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-11.6 - 6.74i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 7.54T + 961T^{2} \)
37 \( 1 + (24.1 + 41.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-43.6 + 25.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (27.3 - 47.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 40.2iT - 2.20e3T^{2} \)
53 \( 1 + (-45.7 - 26.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 56.1iT - 3.48e3T^{2} \)
61 \( 1 - 35.6T + 3.72e3T^{2} \)
67 \( 1 + 62.1T + 4.48e3T^{2} \)
71 \( 1 + 14.5iT - 5.04e3T^{2} \)
73 \( 1 + (-44.6 + 77.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 13.5T + 6.24e3T^{2} \)
83 \( 1 + (33.1 + 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-75.1 + 43.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-11.9 + 20.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.37374400899170582521247391606, −11.60965278449499178852647505192, −10.78012098093301102828198642718, −9.195634430732042856422972770850, −8.760604116507629926800501766387, −7.29668610794417997033124687976, −6.53759694907709005302816316176, −5.64064626784914398667919848633, −4.53808022262166197970293273557, −2.21013255982778939104879596193, 1.26492747594447808665478609942, 2.20980683299815596755424367783, 4.03425434364650307434251247551, 5.04738388713274175605727410260, 6.53424090389366456553228092336, 8.405228735372021456702063724481, 9.172154233580444918785892258026, 10.22288643516407726623155879927, 10.90125782471781532143332804504, 11.83893424287954334251484171230

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