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Properties

Label 2-189-63.59-c3-0-10
Degree $2$
Conductor $189$
Sign $0.997 + 0.0760i$
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  + (−0.725 − 0.418i)2-s + (−3.64 − 6.32i)4-s + 21.9·5-s + (2.19 + 18.3i)7-s + 12.8i·8-s + (−15.8 − 9.17i)10-s + 15.4i·11-s + (33.3 + 19.2i)13-s + (6.10 − 14.2i)14-s + (−23.8 + 41.2i)16-s + (34.8 − 60.3i)17-s + (−55.4 + 32.0i)19-s + (−79.9 − 138. i)20-s + (6.47 − 11.2i)22-s + 69.8i·23-s + ⋯
L(s)  = 1  + (−0.256 − 0.148i)2-s + (−0.456 − 0.790i)4-s + 1.96·5-s + (0.118 + 0.992i)7-s + 0.566i·8-s + (−0.502 − 0.290i)10-s + 0.423i·11-s + (0.711 + 0.410i)13-s + (0.116 − 0.272i)14-s + (−0.372 + 0.644i)16-s + (0.497 − 0.861i)17-s + (−0.669 + 0.386i)19-s + (−0.894 − 1.54i)20-s + (0.0627 − 0.108i)22-s + 0.633i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0760i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 + 0.0760i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.997 + 0.0760i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.95981 - 0.0746217i\)
\(L(\frac12)\) \(\approx\) \(1.95981 - 0.0746217i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.19 - 18.3i)T \)
good2 \( 1 + (0.725 + 0.418i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 21.9T + 125T^{2} \)
11 \( 1 - 15.4iT - 1.33e3T^{2} \)
13 \( 1 + (-33.3 - 19.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-34.8 + 60.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (55.4 - 32.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 69.8iT - 1.21e4T^{2} \)
29 \( 1 + (-168. + 97.3i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-68.0 + 39.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-3.34 - 5.79i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (9.21 - 15.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (12.2 + 21.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-138. + 239. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-95.3 - 55.0i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (176. + 306. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (460. + 265. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (262. + 453. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 43.3iT - 3.57e5T^{2} \)
73 \( 1 + (-54.9 - 31.7i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (606. - 1.04e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (111. + 192. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-35.2 - 61.0i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-483. + 279. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.11542327493295991440511097103, −10.82559768654671474921541954247, −9.863441348665192821221882398944, −9.372119648598168004102876039091, −8.495918803699623401578544575211, −6.45808259144570377411509791634, −5.75740436484252220607903603226, −4.88640714242337290404624858909, −2.44024973932424389843776793480, −1.43901945771518581215800355142, 1.17368490566573577076447724059, 2.98461308836690407279653199023, 4.50693343633733685584909338485, 5.92990023094635417883045720630, 6.85055382149674260997434203271, 8.275507296785120035894522159844, 9.043350841612394335117779976903, 10.20964180536686573987572613237, 10.68951707389055717424539968685, 12.47960638456013593978305386730

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