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Properties

Label 2-189-63.16-c3-0-10
Degree $2$
Conductor $189$
Sign $-0.797 + 0.603i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.54 − 4.41i)2-s + (−8.97 + 15.5i)4-s − 6.84·5-s + (16.1 − 9.05i)7-s + 50.6·8-s + (17.4 + 30.1i)10-s + 41.0·11-s + (31.8 + 55.2i)13-s + (−81.0 − 48.2i)14-s + (−57.3 − 99.2i)16-s + (−38.0 − 65.8i)17-s + (46.7 − 81.0i)19-s + (61.3 − 106. i)20-s + (−104. − 181. i)22-s − 42.1·23-s + ⋯
L(s)  = 1  + (−0.900 − 1.55i)2-s + (−1.12 + 1.94i)4-s − 0.611·5-s + (0.872 − 0.488i)7-s + 2.24·8-s + (0.550 + 0.954i)10-s + 1.12·11-s + (0.680 + 1.17i)13-s + (−1.54 − 0.920i)14-s + (−0.895 − 1.55i)16-s + (−0.542 − 0.939i)17-s + (0.564 − 0.978i)19-s + (0.686 − 1.18i)20-s + (−1.01 − 1.75i)22-s − 0.381·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.797 + 0.603i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.797 + 0.603i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.305853 - 0.911239i\)
\(L(\frac12)\) \(\approx\) \(0.305853 - 0.911239i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.1 + 9.05i)T \)
good2 \( 1 + (2.54 + 4.41i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 6.84T + 125T^{2} \)
11 \( 1 - 41.0T + 1.33e3T^{2} \)
13 \( 1 + (-31.8 - 55.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (38.0 + 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-46.7 + 81.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 42.1T + 1.21e4T^{2} \)
29 \( 1 + (21.9 - 37.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-100. + 173. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (52.1 - 90.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-108. - 188. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-236. + 409. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (8.94 + 15.4i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (211. + 365. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-145. + 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-77.0 - 133. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-419. + 726. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 940.T + 3.57e5T^{2} \)
73 \( 1 + (-65.8 - 113. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (310. + 537. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (102. - 177. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (432. - 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-331. + 573. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.37116704418912393189791055831, −11.19513542717345608033591099068, −9.700921727063569866459881026094, −8.999613727142717838207985507242, −8.040414360818502236255101131214, −6.90746408068782569635475371983, −4.50549613563592472226548472814, −3.71315101139068343953465143498, −2.01322354068019977490942220986, −0.71980563490935591520740512158, 1.21826832445227794862994678482, 4.05560359991825558972642035694, 5.52890574554533950254372687890, 6.32134389282474338525568682528, 7.68280739421240346962723477741, 8.234845707037183895640162543795, 9.038999931875956029535668704174, 10.24289767160448389693517890416, 11.32222414463259103642415538275, 12.48720140405893004820326604176

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