Properties

Label 2-38-19.12-c8-0-3
Degree $2$
Conductor $38$
Sign $0.798 - 0.602i$
Analytic cond. $15.4803$
Root an. cond. $3.93451$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.79 − 5.65i)2-s + (−36.3 + 20.9i)3-s + (63.9 − 110. i)4-s + (106. + 184. i)5-s + (−237. + 411. i)6-s + 680.·7-s − 1.44e3i·8-s + (−2.39e3 + 4.15e3i)9-s + (2.09e3 + 1.20e3i)10-s + 377.·11-s + 5.37e3i·12-s + (3.39e4 + 1.96e4i)13-s + (6.66e3 − 3.84e3i)14-s + (−7.76e3 − 4.48e3i)15-s + (−8.19e3 − 1.41e4i)16-s + (6.22e4 + 1.07e5i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.448 + 0.259i)3-s + (0.249 − 0.433i)4-s + (0.170 + 0.295i)5-s + (−0.183 + 0.317i)6-s + 0.283·7-s − 0.353i·8-s + (−0.365 + 0.633i)9-s + (0.209 + 0.120i)10-s + 0.0257·11-s + 0.259i·12-s + (1.18 + 0.686i)13-s + (0.173 − 0.100i)14-s + (−0.153 − 0.0885i)15-s + (−0.125 − 0.216i)16-s + (0.745 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.798 - 0.602i$
Analytic conductor: \(15.4803\)
Root analytic conductor: \(3.93451\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :4),\ 0.798 - 0.602i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.20573 + 0.738586i\)
\(L(\frac12)\) \(\approx\) \(2.20573 + 0.738586i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.79 + 5.65i)T \)
19 \( 1 + (-1.24e5 - 3.88e4i)T \)
good3 \( 1 + (36.3 - 20.9i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (-106. - 184. i)T + (-1.95e5 + 3.38e5i)T^{2} \)
7 \( 1 - 680.T + 5.76e6T^{2} \)
11 \( 1 - 377.T + 2.14e8T^{2} \)
13 \( 1 + (-3.39e4 - 1.96e4i)T + (4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 + (-6.22e4 - 1.07e5i)T + (-3.48e9 + 6.04e9i)T^{2} \)
23 \( 1 + (9.46e4 - 1.63e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (-9.87e4 - 5.70e4i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 - 1.11e6iT - 8.52e11T^{2} \)
37 \( 1 - 6.61e5iT - 3.51e12T^{2} \)
41 \( 1 + (2.38e6 - 1.37e6i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (1.27e6 + 2.20e6i)T + (-5.84e12 + 1.01e13i)T^{2} \)
47 \( 1 + (-3.25e6 + 5.63e6i)T + (-1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (9.24e6 + 5.33e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (4.80e6 - 2.77e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-3.69e6 + 6.39e6i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (6.67e5 + 3.85e5i)T + (2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + (-9.80e6 + 5.66e6i)T + (3.22e14 - 5.59e14i)T^{2} \)
73 \( 1 + (-2.03e7 - 3.52e7i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-3.48e6 + 2.01e6i)T + (7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 8.37e7T + 2.25e15T^{2} \)
89 \( 1 + (-3.07e7 - 1.77e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + (-1.76e7 + 1.01e7i)T + (3.91e15 - 6.78e15i)T^{2} \)
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

−14.38986372060716060241599814921, −13.61039537509013072352350971827, −12.10122543153128987034460603071, −11.07479772669642779702528512256, −10.14250335488608568951542312364, −8.286668055816663007224313138374, −6.37274412712432994600949740217, −5.16600108033204156728745790347, −3.54708418326789976809577705262, −1.59821184852963462163445806313, 0.893614850052501913530783061257, 3.25556429315091504353038030160, 5.14170770268785663335463023457, 6.22589913566196318748776420400, 7.73627353699018937830580757260, 9.261033527426894249732341314225, 11.12715264385352723302657427106, 12.09443019706743072386363006345, 13.26294345561934307487314022016, 14.32191331491675653274492353047

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