Properties

Label 2-38-19.16-c3-0-3
Degree $2$
Conductor $38$
Sign $0.669 + 0.742i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
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  + (−1.53 − 1.28i)2-s + (9.35 − 3.40i)3-s + (0.694 + 3.93i)4-s + (−0.989 + 5.61i)5-s + (−18.7 − 6.81i)6-s + (−3.91 − 6.78i)7-s + (4.00 − 6.92i)8-s + (55.2 − 46.3i)9-s + (8.73 − 7.32i)10-s + (−17.7 + 30.7i)11-s + (19.9 + 34.4i)12-s + (−58.7 − 21.3i)13-s + (−2.72 + 15.4i)14-s + (9.85 + 55.8i)15-s + (−15.0 + 5.47i)16-s + (47.7 + 40.1i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (1.80 − 0.655i)3-s + (0.0868 + 0.492i)4-s + (−0.0885 + 0.501i)5-s + (−1.27 − 0.463i)6-s + (−0.211 − 0.366i)7-s + (0.176 − 0.306i)8-s + (2.04 − 1.71i)9-s + (0.276 − 0.231i)10-s + (−0.487 + 0.843i)11-s + (0.479 + 0.829i)12-s + (−1.25 − 0.455i)13-s + (−0.0519 + 0.294i)14-s + (0.169 + 0.961i)15-s + (−0.234 + 0.0855i)16-s + (0.681 + 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41485 - 0.629430i\)
\(L(\frac12)\) \(\approx\) \(1.41485 - 0.629430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.53 + 1.28i)T \)
19 \( 1 + (51.9 - 64.5i)T \)
good3 \( 1 + (-9.35 + 3.40i)T + (20.6 - 17.3i)T^{2} \)
5 \( 1 + (0.989 - 5.61i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (3.91 + 6.78i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (17.7 - 30.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (58.7 + 21.3i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (-47.7 - 40.1i)T + (853. + 4.83e3i)T^{2} \)
23 \( 1 + (-19.2 - 108. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (84.0 - 70.5i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (166. + 288. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 36.0T + 5.06e4T^{2} \)
41 \( 1 + (11.1 - 4.05i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (-13.6 + 77.4i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (-179. + 150. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + (-13.5 - 76.8i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (374. + 314. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (24.0 + 136. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-378. + 317. i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (17.6 - 100. i)T + (-3.36e5 - 1.22e5i)T^{2} \)
73 \( 1 + (533. - 194. i)T + (2.98e5 - 2.50e5i)T^{2} \)
79 \( 1 + (-351. + 127. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (57.8 + 100. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-653. - 237. i)T + (5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (-740. - 621. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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

−15.16950628317851097739592426293, −14.60327893109114629235854951790, −13.15776831828721212033225941082, −12.41411294703669351812022349458, −10.28347387594818784329757432950, −9.388018483028110025373408098996, −7.84715755411062860248289401988, −7.25172558007123213777060018599, −3.57338231137355861806962637157, −2.13218745854444005839414104953, 2.71196390235108175370140576676, 4.80377791297991526702420624539, 7.37142913214427039850959071049, 8.618396933284118493580898090382, 9.257794030614238740796329535350, 10.49646123320521623819522225758, 12.72186667944095859347503211184, 14.06431854238656240754293601723, 14.83687009528455165804384683180, 15.92764354454824888294258302664

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