↑

Properties

Label 2-38-19.18-c6-0-6
Degree $2$
Conductor $38$
Sign $0.537 + 0.843i$
Analytic cond. $8.74205$
Root an. cond. $2.95669$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s + 6.65i·3-s − 32.0·4-s + 146.·5-s + 37.6·6-s − 183.·7-s + 181. i·8-s + 684.·9-s − 831. i·10-s + 1.71e3·11-s − 213. i·12-s − 4.16e3i·13-s + 1.03e3i·14-s + 978. i·15-s + 1.02e3·16-s − 5.34e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.246i·3-s − 0.500·4-s + 1.17·5-s + 0.174·6-s − 0.535·7-s + 0.353i·8-s + 0.939·9-s − 0.831i·10-s + 1.29·11-s − 0.123i·12-s − 1.89i·13-s + 0.378i·14-s + 0.289i·15-s + 0.250·16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(8.74205\)
Root analytic conductor: \(2.95669\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3),\ 0.537 + 0.843i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.75635 - 0.963240i\)
\(L(\frac12)\) \(\approx\) \(1.75635 - 0.963240i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
19 \( 1 + (-5.78e3 + 3.68e3i)T \)
good3 \( 1 - 6.65iT - 729T^{2} \)
5 \( 1 - 146.T + 1.56e4T^{2} \)
7 \( 1 + 183.T + 1.17e5T^{2} \)
11 \( 1 - 1.71e3T + 1.77e6T^{2} \)
13 \( 1 + 4.16e3iT - 4.82e6T^{2} \)
17 \( 1 + 5.34e3T + 2.41e7T^{2} \)
23 \( 1 - 1.74e4T + 1.48e8T^{2} \)
29 \( 1 - 2.12e4iT - 5.94e8T^{2} \)
31 \( 1 - 5.17e4iT - 8.87e8T^{2} \)
37 \( 1 + 8.13e4iT - 2.56e9T^{2} \)
41 \( 1 - 9.84e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.55e4T + 6.32e9T^{2} \)
47 \( 1 + 1.72e4T + 1.07e10T^{2} \)
53 \( 1 + 3.96e4iT - 2.21e10T^{2} \)
59 \( 1 + 2.93e5iT - 4.21e10T^{2} \)
61 \( 1 + 8.06e4T + 5.15e10T^{2} \)
67 \( 1 - 3.51e5iT - 9.04e10T^{2} \)
71 \( 1 - 9.89e4iT - 1.28e11T^{2} \)
73 \( 1 + 3.25e5T + 1.51e11T^{2} \)
79 \( 1 + 1.37e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.42e5T + 3.26e11T^{2} \)
89 \( 1 - 3.79e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.29e5iT - 8.32e11T^{2} \)
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

−14.68566924252724091665063218262, −13.30380128971167360699652821506, −12.70010764676545634319516711416, −10.92991800527366117579677416181, −9.886400997675688247568086639234, −9.031889085593824357853059719902, −6.77840863947209273012595571164, −5.11314620554732559185490469663, −3.18887388143159736315617136492, −1.25344850822753627618803157842, 1.63320147756624966373125694287, 4.31739039182338991389965632526, 6.26619633180208426215869370360, 6.96674244493476097729150581914, 9.143373699046579313091231955265, 9.729464475531727766922319194803, 11.73887150526391241147029907530, 13.30733280262766327429204958065, 13.87172095230976500643103970893, 15.20630220472490330862752589081

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