Properties

Label 2-38-19.9-c5-0-2
Degree $2$
Conductor $38$
Sign $0.265 - 0.964i$
Analytic cond. $6.09458$
Root an. cond. $2.46872$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.694 − 3.93i)2-s + (−3.81 + 3.20i)3-s + (−15.0 + 5.47i)4-s + (−0.763 − 0.277i)5-s + (15.2 + 12.8i)6-s + (−52.7 + 91.4i)7-s + (32 + 55.4i)8-s + (−37.8 + 214. i)9-s + (−0.564 + 3.20i)10-s + (145. + 251. i)11-s + (39.8 − 68.9i)12-s + (251. + 211. i)13-s + (396. + 144. i)14-s + (3.80 − 1.38i)15-s + (196. − 164. i)16-s + (102. + 580. i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.244 + 0.205i)3-s + (−0.469 + 0.171i)4-s + (−0.0136 − 0.00497i)5-s + (0.173 + 0.145i)6-s + (−0.407 + 0.705i)7-s + (0.176 + 0.306i)8-s + (−0.155 + 0.884i)9-s + (−0.00178 + 0.0101i)10-s + (0.362 + 0.627i)11-s + (0.0798 − 0.138i)12-s + (0.413 + 0.346i)13-s + (0.541 + 0.196i)14-s + (0.00436 − 0.00158i)15-s + (0.191 − 0.160i)16-s + (0.0859 + 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(6.09458\)
Root analytic conductor: \(2.46872\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.707485 + 0.538851i\)
\(L(\frac12)\) \(\approx\) \(0.707485 + 0.538851i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.694 + 3.93i)T \)
19 \( 1 + (1.52e3 - 372. i)T \)
good3 \( 1 + (3.81 - 3.20i)T + (42.1 - 239. i)T^{2} \)
5 \( 1 + (0.763 + 0.277i)T + (2.39e3 + 2.00e3i)T^{2} \)
7 \( 1 + (52.7 - 91.4i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-145. - 251. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-251. - 211. i)T + (6.44e4 + 3.65e5i)T^{2} \)
17 \( 1 + (-102. - 580. i)T + (-1.33e6 + 4.85e5i)T^{2} \)
23 \( 1 + (1.80e3 - 658. i)T + (4.93e6 - 4.13e6i)T^{2} \)
29 \( 1 + (-47.2 + 267. i)T + (-1.92e7 - 7.01e6i)T^{2} \)
31 \( 1 + (-324. + 561. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 5.82e3T + 6.93e7T^{2} \)
41 \( 1 + (-3.11e3 + 2.61e3i)T + (2.01e7 - 1.14e8i)T^{2} \)
43 \( 1 + (-5.14e3 - 1.87e3i)T + (1.12e8 + 9.44e7i)T^{2} \)
47 \( 1 + (3.53e3 - 2.00e4i)T + (-2.15e8 - 7.84e7i)T^{2} \)
53 \( 1 + (-3.46e3 + 1.26e3i)T + (3.20e8 - 2.68e8i)T^{2} \)
59 \( 1 + (-3.98e3 - 2.25e4i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (-3.22e4 + 1.17e4i)T + (6.46e8 - 5.42e8i)T^{2} \)
67 \( 1 + (-3.58e3 + 2.03e4i)T + (-1.26e9 - 4.61e8i)T^{2} \)
71 \( 1 + (2.07e4 + 7.56e3i)T + (1.38e9 + 1.15e9i)T^{2} \)
73 \( 1 + (-6.15e4 + 5.16e4i)T + (3.59e8 - 2.04e9i)T^{2} \)
79 \( 1 + (5.17e3 - 4.34e3i)T + (5.34e8 - 3.03e9i)T^{2} \)
83 \( 1 + (3.03e4 - 5.25e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (3.22e4 + 2.70e4i)T + (9.69e8 + 5.49e9i)T^{2} \)
97 \( 1 + (-5.91e3 - 3.35e4i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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

−15.62201919926337180616663812441, −14.21317676343903245492639448160, −12.90248186184927882144315279499, −11.85572739480245503788222480506, −10.65642752291488011653638441187, −9.481701848303923506657612264221, −8.104415359177344273958594901204, −6.01329837020320853960396526013, −4.24857806486183891145739008470, −2.15460931006822165391361810212, 0.54705690007341072079143643952, 3.81524684042841007601402615805, 5.90258522650577891631034900995, 6.98031927709979572706742815435, 8.537529275748809701766397357002, 9.881369987311827445991794327564, 11.37561071704154475618008257077, 12.85997519596324352394143641685, 13.94154927882629457084353749346, 15.13204420212008615743115842270

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