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Properties

Label 2-38-19.8-c4-0-1
Degree $2$
Conductor $38$
Sign $0.436 - 0.899i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (−14.7 − 8.53i)3-s + (3.99 + 6.92i)4-s + (−3.60 + 6.25i)5-s + (24.1 + 41.8i)6-s + 27.7·7-s − 22.6i·8-s + (105. + 182. i)9-s + (17.6 − 10.2i)10-s + 43.1·11-s − 136. i·12-s + (−274. + 158. i)13-s + (−68.0 − 39.3i)14-s + (106. − 61.6i)15-s + (−32.0 + 55.4i)16-s + (37.2 − 64.5i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.64 − 0.948i)3-s + (0.249 + 0.433i)4-s + (−0.144 + 0.250i)5-s + (0.670 + 1.16i)6-s + 0.567·7-s − 0.353i·8-s + (1.29 + 2.24i)9-s + (0.176 − 0.102i)10-s + 0.356·11-s − 0.948i·12-s + (−1.62 + 0.938i)13-s + (−0.347 − 0.200i)14-s + (0.474 − 0.273i)15-s + (−0.125 + 0.216i)16-s + (0.128 − 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.436 - 0.899i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.297673 + 0.186396i\)
\(L(\frac12)\) \(\approx\) \(0.297673 + 0.186396i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 + 1.41i)T \)
19 \( 1 + (-172. - 317. i)T \)
good3 \( 1 + (14.7 + 8.53i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (3.60 - 6.25i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 - 27.7T + 2.40e3T^{2} \)
11 \( 1 - 43.1T + 1.46e4T^{2} \)
13 \( 1 + (274. - 158. i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-37.2 + 64.5i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (48.8 + 84.6i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (1.05e3 - 610. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 256. iT - 9.23e5T^{2} \)
37 \( 1 - 969. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.66e3 - 961. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.49e3 - 2.58e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.74e3 + 3.02e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.52e3 + 1.45e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (2.21e3 + 1.27e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.68e3 + 2.91e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (388. - 224. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-3.38e3 - 1.95e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (2.33e3 - 4.04e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-74.9 - 43.2i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 2.58e3T + 4.74e7T^{2} \)
89 \( 1 + (7.55e3 - 4.36e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-9.39e3 - 5.42e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−16.53577346524044238512699320203, −14.52736064497504612291909655004, −12.82423634306029963685477720251, −11.81839742881354830469320373154, −11.24572462995061609545237655401, −9.847815078818591001299146686164, −7.68113619534978048789434983374, −6.75846987908810289210061078630, −5.03371065697883342300944496800, −1.61164480875419282271498598178, 0.36535399924958679547091248142, 4.65634091545453687925683962173, 5.72631948262721441276529756300, 7.37180204452455348620448761211, 9.363531224606936751119820979987, 10.38776629873134148511611956043, 11.44203870473597036140377755277, 12.43098717598241859521173440551, 14.78783811321203683468926215695, 15.61067899178246485351435602471

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