↑

Properties

Label 2-189-63.59-c3-0-8
Degree $2$
Conductor $189$
Sign $0.440 + 0.897i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.52 − 1.45i)2-s + (0.247 + 0.428i)4-s − 17.2·5-s + (17.7 + 5.37i)7-s + 21.8i·8-s + (43.4 + 25.0i)10-s + 35.6i·11-s + (−8.99 − 5.19i)13-s + (−36.9 − 39.3i)14-s + (33.8 − 58.6i)16-s + (64.1 − 111. i)17-s + (−84.7 + 48.9i)19-s + (−4.26 − 7.38i)20-s + (52.0 − 90.0i)22-s − 25.5i·23-s + ⋯
L(s)  = 1  + (−0.892 − 0.515i)2-s + (0.0309 + 0.0536i)4-s − 1.53·5-s + (0.956 + 0.290i)7-s + 0.966i·8-s + (1.37 + 0.793i)10-s + 0.978i·11-s + (−0.191 − 0.110i)13-s + (−0.704 − 0.752i)14-s + (0.529 − 0.916i)16-s + (0.915 − 1.58i)17-s + (−1.02 + 0.590i)19-s + (−0.0476 − 0.0825i)20-s + (0.504 − 0.873i)22-s − 0.231i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.440 + 0.897i$
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: \((2,\ 189,\ (\ :3/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.600272 - 0.374133i\)
\(L(\frac12)\) \(\approx\) \(0.600272 - 0.374133i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.7 - 5.37i)T \)
good2 \( 1 + (2.52 + 1.45i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 17.2T + 125T^{2} \)
11 \( 1 - 35.6iT - 1.33e3T^{2} \)
13 \( 1 + (8.99 + 5.19i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-64.1 + 111. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (84.7 - 48.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 25.5iT - 1.21e4T^{2} \)
29 \( 1 + (-14.1 + 8.16i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-160. + 92.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (0.462 + 0.801i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-231. + 401. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (43.2 + 74.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-143. + 248. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-334. - 193. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-45.2 - 78.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-120. - 69.7i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (151. + 262. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 351. iT - 3.57e5T^{2} \)
73 \( 1 + (-761. - 439. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-7.12 + 12.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-125. - 217. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-505. - 875. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.12e3 + 649. i)T + (4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.88367674221471711462225772987, −10.95305815429150197736674558379, −10.02503395973335109644271908506, −8.865392778632929176885464756919, −7.988828111456014828232858972139, −7.32136197521446065754776701387, −5.25301364740882104571624581946, −4.22701956398614133544807110644, −2.36681828037535990281032776950, −0.64500195164152672274667852789, 0.876708078581715513564664670924, 3.54167244985277091625522184965, 4.50001028503634220515123062507, 6.35579128347416154605891495228, 7.64634916007041852872229050387, 8.138420552684901717660413426376, 8.779642873820960206552793642414, 10.36269387836363646344830574141, 11.18360778898047696291630401487, 12.12548807506303799776691219014

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