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Properties

Label 2-189-63.58-c3-0-4
Degree $2$
Conductor $189$
Sign $-0.883 - 0.467i$
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  − 2.66·2-s − 0.908·4-s + (9.61 + 16.6i)5-s + (−5.55 + 17.6i)7-s + 23.7·8-s + (−25.6 − 44.3i)10-s + (−19.2 + 33.3i)11-s + (38.2 − 66.3i)13-s + (14.7 − 47.0i)14-s − 55.9·16-s + (11.9 + 20.6i)17-s + (−23.6 + 41.0i)19-s + (−8.74 − 15.1i)20-s + (51.3 − 88.8i)22-s + (−6.76 − 11.7i)23-s + ⋯
L(s)  = 1  − 0.941·2-s − 0.113·4-s + (0.860 + 1.49i)5-s + (−0.300 + 0.953i)7-s + 1.04·8-s + (−0.810 − 1.40i)10-s + (−0.528 + 0.914i)11-s + (0.816 − 1.41i)13-s + (0.282 − 0.898i)14-s − 0.873·16-s + (0.170 + 0.294i)17-s + (−0.285 + 0.495i)19-s + (−0.0977 − 0.169i)20-s + (0.497 − 0.861i)22-s + (−0.0613 − 0.106i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.883 - 0.467i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.182373 + 0.734432i\)
\(L(\frac12)\) \(\approx\) \(0.182373 + 0.734432i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.55 - 17.6i)T \)
good2 \( 1 + 2.66T + 8T^{2} \)
5 \( 1 + (-9.61 - 16.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (19.2 - 33.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-38.2 + 66.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-11.9 - 20.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (23.6 - 41.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (6.76 + 11.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-53.5 - 92.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 158.T + 2.97e4T^{2} \)
37 \( 1 + (-23.0 + 39.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (101. - 175. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (42.1 + 72.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 473.T + 1.03e5T^{2} \)
53 \( 1 + (-39.8 - 69.0i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 316.T + 2.05e5T^{2} \)
61 \( 1 + 163.T + 2.26e5T^{2} \)
67 \( 1 - 540.T + 3.00e5T^{2} \)
71 \( 1 + 810.T + 3.57e5T^{2} \)
73 \( 1 + (142. + 246. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 734.T + 4.93e5T^{2} \)
83 \( 1 + (-290. - 503. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-463. + 803. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (413. + 715. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.65928770242196000989536108456, −11.04769214283177223240871171945, −10.27714757635273312461023552333, −9.785305855982136341724476018429, −8.578500283478183718588275392962, −7.57341344969995173480136604085, −6.38330874799364019733987984680, −5.34169326214677346128678547963, −3.23935395993394251882862120638, −1.93355958907765547267491004683, 0.47640343052527601608777974047, 1.58915387605812211474960926698, 4.08508970053546693217285861719, 5.16203288987656203682463878352, 6.55659945590394300488907708697, 8.000912102049941670143426036231, 8.862109870429148559572860199852, 9.455836359031130166796223406078, 10.40527768291737904153339385820, 11.46527814328462062634023786942

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