Properties

Label 2-38-19.17-c7-0-6
Degree $2$
Conductor $38$
Sign $0.226 + 0.973i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 7.87i)2-s + (−9.71 − 8.15i)3-s + (−60.1 − 21.8i)4-s + (424. − 154. i)5-s + (−77.7 + 65.2i)6-s + (847. + 1.46e3i)7-s + (−256 + 443. i)8-s + (−351. − 1.99e3i)9-s + (−627. − 3.55e3i)10-s + (−355. + 615. i)11-s + (405. + 703. i)12-s + (7.81e3 − 6.55e3i)13-s + (1.27e4 − 4.63e3i)14-s + (−5.38e3 − 1.95e3i)15-s + (3.13e3 + 2.63e3i)16-s + (3.78e3 − 2.14e4i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.207 − 0.174i)3-s + (−0.469 − 0.171i)4-s + (1.51 − 0.552i)5-s + (−0.146 + 0.123i)6-s + (0.934 + 1.61i)7-s + (−0.176 + 0.306i)8-s + (−0.160 − 0.912i)9-s + (−0.198 − 1.12i)10-s + (−0.0804 + 0.139i)11-s + (0.0678 + 0.117i)12-s + (0.986 − 0.827i)13-s + (1.24 − 0.451i)14-s + (−0.411 − 0.149i)15-s + (0.191 + 0.160i)16-s + (0.186 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.226 + 0.973i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ 0.226 + 0.973i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.82425 - 1.44826i\)
\(L(\frac12)\) \(\approx\) \(1.82425 - 1.44826i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 7.87i)T \)
19 \( 1 + (2.87e4 + 8.23e3i)T \)
good3 \( 1 + (9.71 + 8.15i)T + (379. + 2.15e3i)T^{2} \)
5 \( 1 + (-424. + 154. i)T + (5.98e4 - 5.02e4i)T^{2} \)
7 \( 1 + (-847. - 1.46e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (355. - 615. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-7.81e3 + 6.55e3i)T + (1.08e7 - 6.17e7i)T^{2} \)
17 \( 1 + (-3.78e3 + 2.14e4i)T + (-3.85e8 - 1.40e8i)T^{2} \)
23 \( 1 + (-4.30e4 - 1.56e4i)T + (2.60e9 + 2.18e9i)T^{2} \)
29 \( 1 + (-9.27e3 - 5.26e4i)T + (-1.62e10 + 5.89e9i)T^{2} \)
31 \( 1 + (3.25e4 + 5.64e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 2.62e5T + 9.49e10T^{2} \)
41 \( 1 + (-1.53e5 - 1.29e5i)T + (3.38e10 + 1.91e11i)T^{2} \)
43 \( 1 + (-4.78e5 + 1.74e5i)T + (2.08e11 - 1.74e11i)T^{2} \)
47 \( 1 + (-1.34e5 - 7.61e5i)T + (-4.76e11 + 1.73e11i)T^{2} \)
53 \( 1 + (-1.20e6 - 4.37e5i)T + (8.99e11 + 7.55e11i)T^{2} \)
59 \( 1 + (-2.93e5 + 1.66e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (1.49e6 + 5.44e5i)T + (2.40e12 + 2.02e12i)T^{2} \)
67 \( 1 + (2.20e5 + 1.24e6i)T + (-5.69e12 + 2.07e12i)T^{2} \)
71 \( 1 + (4.78e6 - 1.74e6i)T + (6.96e12 - 5.84e12i)T^{2} \)
73 \( 1 + (-4.18e6 - 3.51e6i)T + (1.91e12 + 1.08e13i)T^{2} \)
79 \( 1 + (4.40e6 + 3.69e6i)T + (3.33e12 + 1.89e13i)T^{2} \)
83 \( 1 + (-1.52e6 - 2.63e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (6.28e6 - 5.27e6i)T + (7.68e12 - 4.35e13i)T^{2} \)
97 \( 1 + (-7.67e5 + 4.35e6i)T + (-7.59e13 - 2.76e13i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.34680666413218240672040378510, −13.06324844791065563123925448167, −12.22448188437513644935454700285, −10.98708987413533018967003861092, −9.362506457518808146448979119876, −8.706629610280457024061700262059, −5.98320707083274159335892007403, −5.18433288861283468406509252385, −2.58007030736730046971866724821, −1.23306902637769674842907025264, 1.65001154429354452057133269588, 4.26407262410028146390297605570, 5.78438257410530742072529353212, 7.02516782766253445958584989266, 8.524117338138853453006846244373, 10.36027720748348530583766159414, 10.85812410162937530790385256145, 13.34444990802731519986266928039, 13.86282675066518232021140931384, 14.71697381169923526474857191011

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