Properties

Degree $2$
Conductor $38$
Sign $-0.992 - 0.119i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 + 2.57i)2-s + (−10.3 + 3.77i)3-s + (2.77 + 15.7i)4-s + (−9.06 + 51.4i)5-s + (−41.5 − 15.1i)6-s + (−107. − 186. i)7-s + (−32.0 + 55.4i)8-s + (−92.6 + 77.7i)9-s + (−159. + 134. i)10-s + (−273. + 473. i)11-s + (−88.3 − 153. i)12-s + (−242. − 88.1i)13-s + (149. − 849. i)14-s + (−100. − 567. i)15-s + (−240. + 87.5i)16-s + (1.45e3 + 1.21e3i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.665 + 0.242i)3-s + (0.0868 + 0.492i)4-s + (−0.162 + 0.919i)5-s + (−0.470 − 0.171i)6-s + (−0.831 − 1.44i)7-s + (−0.176 + 0.306i)8-s + (−0.381 + 0.320i)9-s + (−0.505 + 0.424i)10-s + (−0.681 + 1.18i)11-s + (−0.177 − 0.306i)12-s + (−0.397 − 0.144i)13-s + (0.204 − 1.15i)14-s + (−0.114 − 0.651i)15-s + (−0.234 + 0.0855i)16-s + (1.21 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.992 - 0.119i$
Motivic weight: \(5\)
Character: $\chi_{38} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.992 - 0.119i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0502148 + 0.835274i\)
\(L(\frac12)\) \(\approx\) \(0.0502148 + 0.835274i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 - 2.57i)T \)
19 \( 1 + (61.8 + 1.57e3i)T \)
good3 \( 1 + (10.3 - 3.77i)T + (186. - 156. i)T^{2} \)
5 \( 1 + (9.06 - 51.4i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (107. + 186. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (273. - 473. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (242. + 88.1i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (-1.45e3 - 1.21e3i)T + (2.46e5 + 1.39e6i)T^{2} \)
23 \( 1 + (-387. - 2.19e3i)T + (-6.04e6 + 2.20e6i)T^{2} \)
29 \( 1 + (2.74e3 - 2.30e3i)T + (3.56e6 - 2.01e7i)T^{2} \)
31 \( 1 + (-4.22e3 - 7.31e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 3.25e3T + 6.93e7T^{2} \)
41 \( 1 + (8.41e3 - 3.06e3i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (-3.13e3 + 1.77e4i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (1.55e4 - 1.30e4i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 + (-623. - 3.53e3i)T + (-3.92e8 + 1.43e8i)T^{2} \)
59 \( 1 + (2.35e3 + 1.97e3i)T + (1.24e8 + 7.04e8i)T^{2} \)
61 \( 1 + (1.00e3 + 5.68e3i)T + (-7.93e8 + 2.88e8i)T^{2} \)
67 \( 1 + (-2.96e4 + 2.48e4i)T + (2.34e8 - 1.32e9i)T^{2} \)
71 \( 1 + (-7.39e3 + 4.19e4i)T + (-1.69e9 - 6.17e8i)T^{2} \)
73 \( 1 + (4.16e4 - 1.51e4i)T + (1.58e9 - 1.33e9i)T^{2} \)
79 \( 1 + (1.03e4 - 3.77e3i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (-1.07e4 - 1.86e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-5.77e4 - 2.10e4i)T + (4.27e9 + 3.58e9i)T^{2} \)
97 \( 1 + (-1.20e5 - 1.01e5i)T + (1.49e9 + 8.45e9i)T^{2} \)
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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

−15.82135959388373227282202782591, −14.72161085832319710110389904928, −13.58203708029576193182488995570, −12.40508300321167631590137961408, −10.82449546999918320338963578066, −10.11517935268577134129193111144, −7.58246604171922092637423777701, −6.70597028517568969071017207841, −5.02127692998005472979493209725, −3.32157586782686626365494959596, 0.41985586360217012111305605205, 2.96616632955628662328762753684, 5.29712827482900717237399711419, 6.06622312557186142736599703321, 8.466703615485519757812450864360, 9.760205215921305060161170724689, 11.59663104569121870283417502706, 12.21607083327526370772762553366, 13.10360016607521877222507076484, 14.65477376744031859198801677831

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