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Properties

Label 2-189-63.41-c3-0-11
Degree $2$
Conductor $189$
Sign $0.872 - 0.488i$
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  + (3.38 − 1.95i)2-s + (3.63 − 6.29i)4-s + (−5.82 + 10.0i)5-s + (−2.49 + 18.3i)7-s + 2.86i·8-s + 45.5i·10-s + (41.2 − 23.8i)11-s + (60.2 + 34.7i)13-s + (27.3 + 66.9i)14-s + (34.6 + 60.0i)16-s − 25.6·17-s + 41.1i·19-s + (42.3 + 73.3i)20-s + (93.1 − 161. i)22-s + (−145. − 84.0i)23-s + ⋯
L(s)  = 1  + (1.19 − 0.690i)2-s + (0.454 − 0.786i)4-s + (−0.521 + 0.902i)5-s + (−0.134 + 0.990i)7-s + 0.126i·8-s + 1.43i·10-s + (1.13 − 0.653i)11-s + (1.28 + 0.741i)13-s + (0.523 + 1.27i)14-s + (0.541 + 0.938i)16-s − 0.366·17-s + 0.496i·19-s + (0.473 + 0.820i)20-s + (0.902 − 1.56i)22-s + (−1.32 − 0.762i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.85936 + 0.745345i\)
\(L(\frac12)\) \(\approx\) \(2.85936 + 0.745345i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.49 - 18.3i)T \)
good2 \( 1 + (-3.38 + 1.95i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (5.82 - 10.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-41.2 + 23.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-60.2 - 34.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 25.6T + 4.91e3T^{2} \)
19 \( 1 - 41.1iT - 6.85e3T^{2} \)
23 \( 1 + (145. + 84.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (28.2 - 16.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (148. + 85.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 3.86T + 5.06e4T^{2} \)
41 \( 1 + (-168. + 292. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-207. - 359. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-74.3 - 128. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 59.7iT - 1.48e5T^{2} \)
59 \( 1 + (-282. + 488. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-571. + 330. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-145. + 252. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 3.05iT - 3.57e5T^{2} \)
73 \( 1 + 506. iT - 3.89e5T^{2} \)
79 \( 1 + (96.6 + 167. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (216. + 374. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 924.T + 7.04e5T^{2} \)
97 \( 1 + (943. - 544. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.08800558752095462799582865456, −11.39220361708433052643043714915, −10.86027369828663214308849947229, −9.216774591619926073470799434634, −8.219872097689027276291059457625, −6.49206740041142090106935964268, −5.83388444830969141844611910503, −4.10789151963789298996394578446, −3.41907320537888837904823697048, −2.02919657502216753542583963297, 0.982106734341480672375192157845, 3.80118073231748383789504480142, 4.22796622244995322604848717282, 5.54806565439627639419808039163, 6.66710152435668075158646217818, 7.61793764923467846076252742628, 8.827653160032280083959105583016, 10.05700669094479501564833422716, 11.39156629990000673461735403988, 12.39507350806214988939113755079

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