Properties

Label 2-38-19.11-c9-0-5
Degree $2$
Conductor $38$
Sign $0.336 - 0.941i$
Analytic cond. $19.5713$
Root an. cond. $4.42395$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 13.8i)2-s + (−113. + 196. i)3-s + (−127. − 221. i)4-s + (563. − 975. i)5-s + (1.81e3 + 3.13e3i)6-s + 3.88e3·7-s − 4.09e3·8-s + (−1.58e4 − 2.73e4i)9-s + (−9.00e3 − 1.56e4i)10-s + 1.55e4·11-s + 5.79e4·12-s + (3.17e4 + 5.49e4i)13-s + (3.10e4 − 5.38e4i)14-s + (1.27e5 + 2.20e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (−1.94e5 + 3.36e5i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.807 + 1.39i)3-s + (−0.249 − 0.433i)4-s + (0.402 − 0.697i)5-s + (0.570 + 0.988i)6-s + 0.611·7-s − 0.353·8-s + (−0.803 − 1.39i)9-s + (−0.284 − 0.493i)10-s + 0.319·11-s + 0.807·12-s + (0.307 + 0.533i)13-s + (0.216 − 0.374i)14-s + (0.650 + 1.12i)15-s + (−0.125 + 0.216i)16-s + (−0.563 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 0.336 - 0.941i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.30217 + 0.917188i\)
\(L(\frac12)\) \(\approx\) \(1.30217 + 0.917188i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8 + 13.8i)T \)
19 \( 1 + (2.99e5 - 4.82e5i)T \)
good3 \( 1 + (113. - 196. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-563. + 975. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 - 3.88e3T + 4.03e7T^{2} \)
11 \( 1 - 1.55e4T + 2.35e9T^{2} \)
13 \( 1 + (-3.17e4 - 5.49e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (1.94e5 - 3.36e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
23 \( 1 + (-1.03e6 - 1.80e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + (8.77e5 + 1.51e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 - 5.35e6T + 2.64e13T^{2} \)
37 \( 1 + 3.73e6T + 1.29e14T^{2} \)
41 \( 1 + (5.25e6 - 9.09e6i)T + (-1.63e14 - 2.83e14i)T^{2} \)
43 \( 1 + (1.21e6 - 2.09e6i)T + (-2.51e14 - 4.35e14i)T^{2} \)
47 \( 1 + (-7.75e6 - 1.34e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (-4.64e7 - 8.04e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-7.63e7 + 1.32e8i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-4.07e7 - 7.06e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (1.20e7 + 2.09e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + (4.31e7 - 7.47e7i)T + (-2.29e16 - 3.97e16i)T^{2} \)
73 \( 1 + (2.08e8 - 3.60e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-1.95e8 + 3.38e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 5.21e8T + 1.86e17T^{2} \)
89 \( 1 + (5.36e8 + 9.30e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + (-2.40e8 + 4.17e8i)T + (-3.80e17 - 6.58e17i)T^{2} \)
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

−14.64754762196713962377362433793, −13.20333199217273528046890941287, −11.78914004070618712416937558879, −10.94132955254618187643545028744, −9.834159356556104142149222489938, −8.758525269161363200909371299808, −5.97541340011280268992376773591, −4.86464397385123790125885907707, −3.88227591958802213591214361702, −1.44027023573525054588409789926, 0.61641145819741989410479122217, 2.44014324148002376447540416119, 5.01522818018929235958347178919, 6.44027143722499266501785675001, 7.06009489393476786928067147328, 8.517998376331559554982546826820, 10.76462100548289559159522911899, 11.83278347612413245057873666082, 13.02923615530214531508237807225, 13.90013099818518550982592225211

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