Properties

Label 2-19-19.6-c3-0-3
Degree $2$
Conductor $19$
Sign $0.635 + 0.772i$
Analytic cond. $1.12103$
Root an. cond. $1.05879$
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  + (2.98 − 2.50i)2-s + (−3.72 − 1.35i)3-s + (1.25 − 7.09i)4-s + (2.58 + 14.6i)5-s + (−14.5 + 5.28i)6-s + (5.35 − 9.26i)7-s + (1.55 + 2.68i)8-s + (−8.65 − 7.25i)9-s + (44.5 + 37.3i)10-s + (−14.6 − 25.3i)11-s + (−14.2 + 24.7i)12-s + (−76.4 + 27.8i)13-s + (−7.24 − 41.0i)14-s + (10.2 − 58.1i)15-s + (65.5 + 23.8i)16-s + (61.3 − 51.4i)17-s + ⋯
L(s)  = 1  + (1.05 − 0.886i)2-s + (−0.716 − 0.260i)3-s + (0.156 − 0.886i)4-s + (0.231 + 1.31i)5-s + (−0.988 + 0.359i)6-s + (0.288 − 0.500i)7-s + (0.0686 + 0.118i)8-s + (−0.320 − 0.268i)9-s + (1.40 + 1.18i)10-s + (−0.400 − 0.694i)11-s + (−0.343 + 0.594i)12-s + (−1.63 + 0.593i)13-s + (−0.138 − 0.784i)14-s + (0.176 − 1.00i)15-s + (1.02 + 0.372i)16-s + (0.875 − 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28278 - 0.605611i\)
\(L(\frac12)\) \(\approx\) \(1.28278 - 0.605611i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-81.6 - 13.9i)T \)
good2 \( 1 + (-2.98 + 2.50i)T + (1.38 - 7.87i)T^{2} \)
3 \( 1 + (3.72 + 1.35i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (-2.58 - 14.6i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (-5.35 + 9.26i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (14.6 + 25.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (76.4 - 27.8i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (-61.3 + 51.4i)T + (853. - 4.83e3i)T^{2} \)
23 \( 1 + (-11.6 + 66.1i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-1.78 - 1.49i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (68.9 - 119. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 305.T + 5.06e4T^{2} \)
41 \( 1 + (-213. - 77.8i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (29.9 + 169. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (324. + 271. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + (28.7 - 163. i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (-355. + 298. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (106. - 603. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-166. - 139. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (9.51 + 53.9i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 + (-130. - 47.5i)T + (2.98e5 + 2.50e5i)T^{2} \)
79 \( 1 + (436. + 158. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-43.2 + 74.9i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (443. - 161. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (326. - 274. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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

−18.00466338729526964985198957725, −16.75946964908708131000956605644, −14.51319666273638491321037655863, −14.02188932938363940778153300763, −12.24264207395057901899771343049, −11.35883341886865030819417442986, −10.26731175785302080651057257117, −7.15453061456075717790826901700, −5.32963370730844641234529717070, −3.00514936460027544850512638055, 4.99145201442911442842822967994, 5.46242042914934957657555255337, 7.76122188371221757832566840807, 9.860529537245499824275815959329, 12.08181958564353123064620529743, 12.90512838208524406267470363636, 14.46989990067033071365975545149, 15.65559748828723925529755893317, 16.74227046724546768563905782664, 17.50957168605153546237843705775

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