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Properties

Label 2-189-63.4-c3-0-8
Degree $2$
Conductor $189$
Sign $0.558 - 0.829i$
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  + (0.267 − 0.462i)2-s + (3.85 + 6.68i)4-s + 1.39·5-s + (16.9 + 7.37i)7-s + 8.39·8-s + (0.372 − 0.644i)10-s − 19.3·11-s + (4.05 − 7.03i)13-s + (7.95 − 5.88i)14-s + (−28.6 + 49.5i)16-s + (−28.8 + 49.9i)17-s + (35.7 + 61.9i)19-s + (5.37 + 9.30i)20-s + (−5.18 + 8.97i)22-s + 210.·23-s + ⋯
L(s)  = 1  + (0.0944 − 0.163i)2-s + (0.482 + 0.835i)4-s + 0.124·5-s + (0.917 + 0.398i)7-s + 0.371·8-s + (0.0117 − 0.0203i)10-s − 0.531·11-s + (0.0866 − 0.150i)13-s + (0.151 − 0.112i)14-s + (−0.447 + 0.774i)16-s + (−0.411 + 0.713i)17-s + (0.432 + 0.748i)19-s + (0.0600 + 0.104i)20-s + (−0.0502 + 0.0869i)22-s + 1.90·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.558 - 0.829i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.93051 + 1.02822i\)
\(L(\frac12)\) \(\approx\) \(1.93051 + 1.02822i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.9 - 7.37i)T \)
good2 \( 1 + (-0.267 + 0.462i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 1.39T + 125T^{2} \)
11 \( 1 + 19.3T + 1.33e3T^{2} \)
13 \( 1 + (-4.05 + 7.03i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (28.8 - 49.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.7 - 61.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 210.T + 1.21e4T^{2} \)
29 \( 1 + (-78.1 - 135. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (55.5 + 96.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-26.2 - 45.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-201. + 349. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-89.7 - 155. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (46.4 - 80.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (214. - 371. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (194. + 337. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-176. + 305. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (431. + 746. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 377.T + 3.57e5T^{2} \)
73 \( 1 + (-183. + 318. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-154. + 268. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (110. + 191. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (712. + 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (288. + 500. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.30636199120806178105551987545, −11.24752139794019087852284044702, −10.65743983727505489942933893743, −9.093808097279224858119640355759, −8.111206625181402798100121855070, −7.32888517623729787570756116917, −5.89235098460497653735885364762, −4.59591563477145440187665325535, −3.14738584765350344365128275272, −1.79526028807621037296890295289, 1.01352084134922993111526760146, 2.54449392561861631891021412921, 4.61575633127343413694192402920, 5.44364862062078960066070559049, 6.80738061399230006790571166495, 7.64083171454746949224298245832, 9.024673195951811645391558685590, 10.09781453737029009780781235438, 11.10135638878512952301385529878, 11.57346720347060008164124295311

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