Properties

Degree $2$
Conductor $38$
Sign $-0.999 - 0.0362i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.87 − 0.684i)2-s + (−1.54 − 8.73i)3-s + (3.06 + 2.57i)4-s + (−15.3 + 12.8i)5-s + (−3.08 + 17.4i)6-s + (4.71 − 8.16i)7-s + (−4.00 − 6.92i)8-s + (−48.6 + 17.6i)9-s + (37.6 − 13.7i)10-s + (−24.7 − 42.8i)11-s + (17.7 − 30.7i)12-s + (3.93 − 22.2i)13-s + (−14.4 + 12.1i)14-s + (136. + 114. i)15-s + (2.77 + 15.7i)16-s + (30.4 + 11.0i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.296 − 1.68i)3-s + (0.383 + 0.321i)4-s + (−1.37 + 1.15i)5-s + (−0.209 + 1.18i)6-s + (0.254 − 0.440i)7-s + (−0.176 − 0.306i)8-s + (−1.80 + 0.655i)9-s + (1.19 − 0.433i)10-s + (−0.678 − 1.17i)11-s + (0.426 − 0.739i)12-s + (0.0838 − 0.475i)13-s + (−0.275 + 0.231i)14-s + (2.34 + 1.96i)15-s + (0.0434 + 0.246i)16-s + (0.433 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0362i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.999 - 0.0362i$
Motivic weight: \(3\)
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ -0.999 - 0.0362i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00741046 + 0.408408i\)
\(L(\frac12)\) \(\approx\) \(0.00741046 + 0.408408i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 + 0.684i)T \)
19 \( 1 + (28.5 + 77.7i)T \)
good3 \( 1 + (1.54 + 8.73i)T + (-25.3 + 9.23i)T^{2} \)
5 \( 1 + (15.3 - 12.8i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-4.71 + 8.16i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (24.7 + 42.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-3.93 + 22.2i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-30.4 - 11.0i)T + (3.76e3 + 3.15e3i)T^{2} \)
23 \( 1 + (0.237 + 0.199i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (39.4 - 14.3i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (0.938 - 1.62i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 331.T + 5.06e4T^{2} \)
41 \( 1 + (8.17 + 46.3i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (293. - 246. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-115. + 41.8i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + (71.6 + 60.1i)T + (2.58e4 + 1.46e5i)T^{2} \)
59 \( 1 + (517. + 188. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (404. + 339. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-605. + 220. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-468. + 393. i)T + (6.21e4 - 3.52e5i)T^{2} \)
73 \( 1 + (70.7 + 401. i)T + (-3.65e5 + 1.33e5i)T^{2} \)
79 \( 1 + (-70.3 - 398. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-387. + 670. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (152. - 867. i)T + (-6.62e5 - 2.41e5i)T^{2} \)
97 \( 1 + (1.39e3 + 506. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−15.27660093971354751035670543221, −13.84179021008235631464473251781, −12.59732338445204834083391365249, −11.33793469193301553990783145756, −10.88434535513394287643412345495, −8.137207787255124698628379173981, −7.60087528772962169686226173249, −6.42188507408531512248653071116, −2.98592509311963802813407473184, −0.43792185813842336477761371441, 4.16375678502870495109587636081, 5.24613765911295703480756290836, 7.87174464336526532440507462393, 8.977522192252362195981142438043, 10.05749069955550555830910475570, 11.39408511937068931824352893986, 12.34238131583001225213224146234, 14.91422843179725696230216888411, 15.47654804751090382968870434589, 16.36292483233702017368801206525

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