Properties

Degree $2$
Conductor $38$
Sign $0.976 + 0.216i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s + 6.66i·3-s − 8.00·4-s + 26.7·5-s + 18.8·6-s + 51.4·7-s + 22.6i·8-s + 36.5·9-s − 75.5i·10-s − 25.0·11-s − 53.3i·12-s − 53.2i·13-s − 145. i·14-s + 177. i·15-s + 64.0·16-s − 24.4·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.740i·3-s − 0.500·4-s + 1.06·5-s + 0.523·6-s + 1.04·7-s + 0.353i·8-s + 0.451·9-s − 0.755i·10-s − 0.207·11-s − 0.370i·12-s − 0.315i·13-s − 0.742i·14-s + 0.790i·15-s + 0.250·16-s − 0.0847·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.976 + 0.216i$
Motivic weight: \(4\)
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.71647 - 0.187677i\)
\(L(\frac12)\) \(\approx\) \(1.71647 - 0.187677i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
19 \( 1 + (78.0 - 352. i)T \)
good3 \( 1 - 6.66iT - 81T^{2} \)
5 \( 1 - 26.7T + 625T^{2} \)
7 \( 1 - 51.4T + 2.40e3T^{2} \)
11 \( 1 + 25.0T + 1.46e4T^{2} \)
13 \( 1 + 53.2iT - 2.85e4T^{2} \)
17 \( 1 + 24.4T + 8.35e4T^{2} \)
23 \( 1 + 612.T + 2.79e5T^{2} \)
29 \( 1 + 1.34e3iT - 7.07e5T^{2} \)
31 \( 1 + 912. iT - 9.23e5T^{2} \)
37 \( 1 - 325. iT - 1.87e6T^{2} \)
41 \( 1 + 51.7iT - 2.82e6T^{2} \)
43 \( 1 + 2.54e3T + 3.41e6T^{2} \)
47 \( 1 - 2.99e3T + 4.87e6T^{2} \)
53 \( 1 + 4.06e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.07e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.53e3T + 1.38e7T^{2} \)
67 \( 1 - 7.38e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.80e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.62e3T + 2.83e7T^{2} \)
79 \( 1 - 3.90e3iT - 3.89e7T^{2} \)
83 \( 1 + 997.T + 4.74e7T^{2} \)
89 \( 1 + 6.15e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.02e3iT - 8.85e7T^{2} \)
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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

−15.35500712429809092029199780087, −14.21954420427708587744937818414, −13.17514953180490120753543262493, −11.69436680029989981929521937821, −10.35113653537174869409688118694, −9.719887691484818808938212571192, −8.106606854627608753352867610216, −5.67503612724174396467737138668, −4.22969103123084269542977091926, −1.91320333518087895849646334599, 1.71865712054477534848608842638, 4.90278258189868764941967809154, 6.40372248941917225152295842607, 7.64427720294244875826383519386, 9.043788735028684093871084508381, 10.54799251141915194390078115030, 12.28687928141224819961554708223, 13.53329645945474892025726083849, 14.19660766043483425472678564884, 15.54300543256693676829556767914

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