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Properties

Label 2-38-19.12-c2-0-0
Degree $2$
Conductor $38$
Sign $0.999 - 0.0186i$
Analytic cond. $1.03542$
Root an. cond. $1.01755$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (2.72 − 1.57i)3-s + (0.999 − 1.73i)4-s + (0.5 + 0.866i)5-s + (−2.22 + 3.85i)6-s + 6.89·7-s + 2.82i·8-s + (0.449 − 0.778i)9-s + (−1.22 − 0.707i)10-s − 14.8·11-s − 6.29i·12-s + (−14.8 − 8.57i)13-s + (−8.44 + 4.87i)14-s + (2.72 + 1.57i)15-s + (−2.00 − 3.46i)16-s + (1.05 + 1.81i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.908 − 0.524i)3-s + (0.249 − 0.433i)4-s + (0.100 + 0.173i)5-s + (−0.370 + 0.642i)6-s + 0.985·7-s + 0.353i·8-s + (0.0499 − 0.0865i)9-s + (−0.122 − 0.0707i)10-s − 1.35·11-s − 0.524i·12-s + (−1.14 − 0.659i)13-s + (−0.603 + 0.348i)14-s + (0.181 + 0.104i)15-s + (−0.125 − 0.216i)16-s + (0.0617 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.999 - 0.0186i$
Analytic conductor: \(1.03542\)
Root analytic conductor: \(1.01755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1),\ 0.999 - 0.0186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03333 + 0.00962280i\)
\(L(\frac12)\) \(\approx\) \(1.03333 + 0.00962280i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
19 \( 1 + (-11.3 - 15.2i)T \)
good3 \( 1 + (-2.72 + 1.57i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 6.89T + 49T^{2} \)
11 \( 1 + 14.8T + 121T^{2} \)
13 \( 1 + (14.8 + 8.57i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-1.05 - 1.81i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (13.5 - 23.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (5.54 + 3.20i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + 28.9iT - 1.36e3T^{2} \)
41 \( 1 + (-55.9 + 32.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-37.6 - 65.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-5.77 + 9.99i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (69.2 + 40.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-50.9 + 29.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.09 - 1.89i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (51.6 + 29.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-87.5 + 50.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-63.6 - 110. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (5.78 - 3.33i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 1.30T + 6.88e3T^{2} \)
89 \( 1 + (-5.84 - 3.37i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (114. - 65.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−16.03459962440180201716451942219, −14.78746843797827220182340668075, −14.06344531084738997248712123024, −12.68274437523252759675381058944, −10.98813343730996135539579348672, −9.694469557230542856515079575397, −7.896296860673211581392400286071, −7.76807031599980216990108717716, −5.39076458835976442625096338054, −2.37116933867081406711314063484, 2.62235909341781316740940229267, 4.79582525280075482417529520426, 7.50918741324543843252342036686, 8.644611039764738032815085945508, 9.718781360420829958507116799836, 10.97053839799199521273908574942, 12.36797922189140491900020037810, 13.92826031906475871467924960364, 14.93043873439909996026628427657, 16.04514637616901153938514640284

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