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Properties

Label 2-189-63.38-c3-0-6
Degree $2$
Conductor $189$
Sign $-0.680 - 0.732i$
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  + 0.257i·2-s + 7.93·4-s + (−3.19 + 5.53i)5-s + (−15.2 + 10.5i)7-s + 4.10i·8-s + (−1.42 − 0.822i)10-s + (−52.8 + 30.5i)11-s + (−8.78 + 5.07i)13-s + (−2.72 − 3.91i)14-s + 62.4·16-s + (22.5 − 38.9i)17-s + (−69.6 + 40.2i)19-s + (−25.3 + 43.8i)20-s + (−7.86 − 13.6i)22-s + (−23.9 − 13.8i)23-s + ⋯
L(s)  = 1  + 0.0910i·2-s + 0.991·4-s + (−0.285 + 0.494i)5-s + (−0.821 + 0.570i)7-s + 0.181i·8-s + (−0.0450 − 0.0260i)10-s + (−1.44 + 0.837i)11-s + (−0.187 + 0.108i)13-s + (−0.0519 − 0.0748i)14-s + 0.975·16-s + (0.321 − 0.556i)17-s + (−0.840 + 0.485i)19-s + (−0.283 + 0.490i)20-s + (−0.0762 − 0.132i)22-s + (−0.217 − 0.125i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.680 - 0.732i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.680 - 0.732i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.460661 + 1.05629i\)
\(L(\frac12)\) \(\approx\) \(0.460661 + 1.05629i\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.2 - 10.5i)T \)
good2 \( 1 - 0.257iT - 8T^{2} \)
5 \( 1 + (3.19 - 5.53i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (52.8 - 30.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (8.78 - 5.07i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-22.5 + 38.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (69.6 - 40.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (23.9 + 13.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (48.9 + 28.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 106. iT - 2.97e4T^{2} \)
37 \( 1 + (-95.6 - 165. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (15.0 + 26.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (185. - 320. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 496.T + 1.03e5T^{2} \)
53 \( 1 + (-601. - 347. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 635.T + 2.05e5T^{2} \)
61 \( 1 + 747. iT - 2.26e5T^{2} \)
67 \( 1 - 164.T + 3.00e5T^{2} \)
71 \( 1 - 278. iT - 3.57e5T^{2} \)
73 \( 1 + (313. + 181. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 557.T + 4.93e5T^{2} \)
83 \( 1 + (-514. + 891. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-730. - 1.26e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (878. + 507. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.43666194948587568487011121421, −11.51655080056156659107173913092, −10.47167797878479123945939181430, −9.770141101452916616407106074992, −8.184750523717583023926739452277, −7.24586325936593278466762560669, −6.37025071537304855640094101395, −5.15064535188439506813485425482, −3.22025183214830782033823286005, −2.25128127367664516320939872933, 0.45779118916002157678981616192, 2.48812216154490971208401571657, 3.74885918605991908197558756360, 5.43428384048858328800070716883, 6.53408997168459193463525814826, 7.63236812461391611266135054929, 8.540408878773859677953558205415, 10.11555881911088862953395786318, 10.66531459757995229830009167287, 11.71955838946370581900259263374

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