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Properties

Label 2-189-63.5-c3-0-8
Degree $2$
Conductor $189$
Sign $0.0437 - 0.999i$
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  + 0.837i·2-s + 7.29·4-s + (10.9 + 18.9i)5-s + (14.8 + 11.0i)7-s + 12.8i·8-s + (−15.8 + 9.17i)10-s + (−13.3 − 7.73i)11-s + (−33.3 − 19.2i)13-s + (−9.29 + 12.4i)14-s + 47.6·16-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 + (60.4 − 34.9i)23-s + ⋯
L(s)  = 1  + 0.296i·2-s + 0.912·4-s + (0.980 + 1.69i)5-s + (0.800 + 0.599i)7-s + 0.566i·8-s + (−0.502 + 0.290i)10-s + (−0.367 − 0.211i)11-s + (−0.711 − 0.410i)13-s + (−0.177 + 0.237i)14-s + 0.744·16-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 + (0.548 − 0.316i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.0437 - 0.999i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.0437 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.84627 + 1.76719i\)
\(L(\frac12)\) \(\approx\) \(1.84627 + 1.76719i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.8 - 11.0i)T \)
good2 \( 1 - 0.837iT - 8T^{2} \)
5 \( 1 + (-10.9 - 18.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (13.3 + 7.73i)T + (665.5 + 1.15e3i)T^{2} \)
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 \( 1 + (-60.4 + 34.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-168. + 97.3i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 78.6iT - 2.97e4T^{2} \)
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 \( 1 - 276.T + 1.03e5T^{2} \)
53 \( 1 + (95.3 - 55.0i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 + 531. iT - 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 - 43.3iT - 3.57e5T^{2} \)
73 \( 1 + (-54.9 + 31.7i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.21e3T + 4.93e5T^{2} \)
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

−12.13730083054433021588708164102, −11.06859535244435825691276932092, −10.68476476141261881146760330388, −9.527165932279243079466184825307, −8.005550612993580920311439156815, −7.01755905895540987169827996797, −6.23922121709092282943921693225, −5.18949292565765085655689849389, −2.77351035684235420654831714349, −2.26372572890433322605494392247, 1.21404594503874440949025942150, 2.14051282012574297793920842146, 4.34198159340026604337818306465, 5.31234895700394057720772565253, 6.57473827446195171374804395540, 7.909866301977739647236660970227, 8.875066286390142465980896701435, 10.03580714543161690670007657094, 10.79245923343635248514758170134, 12.08610004853997373958738171018

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