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Properties

Label 2-38-19.12-c4-0-5
Degree $2$
Conductor $38$
Sign $0.683 + 0.730i$
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 + (10.4 − 6.05i)3-s + (3.99 − 6.92i)4-s + (9.13 + 15.8i)5-s + (17.1 − 29.6i)6-s − 40.6·7-s − 22.6i·8-s + (32.7 − 56.6i)9-s + (44.7 + 25.8i)10-s − 52.4·11-s − 96.8i·12-s + (133. + 77.0i)13-s + (−99.6 + 57.5i)14-s + (191. + 110. i)15-s + (−32.0 − 55.4i)16-s + (−137. − 238. i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.16 − 0.672i)3-s + (0.249 − 0.433i)4-s + (0.365 + 0.632i)5-s + (0.475 − 0.823i)6-s − 0.830·7-s − 0.353i·8-s + (0.404 − 0.699i)9-s + (0.447 + 0.258i)10-s − 0.433·11-s − 0.672i·12-s + (0.789 + 0.455i)13-s + (−0.508 + 0.293i)14-s + (0.850 + 0.491i)15-s + (−0.125 − 0.216i)16-s + (−0.476 − 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.683 + 0.730i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.40058 - 1.04104i\)
\(L(\frac12)\) \(\approx\) \(2.40058 - 1.04104i\)
\(L(3)\) 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 \( 1 + (70.6 - 354. i)T \)
good3 \( 1 + (-10.4 + 6.05i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-9.13 - 15.8i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 40.6T + 2.40e3T^{2} \)
11 \( 1 + 52.4T + 1.46e4T^{2} \)
13 \( 1 + (-133. - 77.0i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (137. + 238. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (448. - 776. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-268. - 155. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 904. iT - 9.23e5T^{2} \)
37 \( 1 - 115. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.83e3 + 1.06e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.00e3 + 1.73e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.01e3 + 3.48e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.55e3 - 1.47e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (5.20e3 - 3.00e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-606. + 1.05e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (5.55e3 + 3.20e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (2.81e3 - 1.62e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.82e3 + 6.62e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-5.80e3 + 3.35e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 8.74e3T + 4.74e7T^{2} \)
89 \( 1 + (-6.84e3 - 3.95e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-6.08e3 + 3.51e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.03566496805535249829826844234, −13.71771253713388018536902309068, −13.54743434579594690595726723433, −12.06839940711854939913488756743, −10.44685098027089923171972721340, −9.125469002049395829519125474752, −7.46655702825951414502970502375, −6.11521657448065452495815955416, −3.52595662590063549766314247511, −2.19478341048614662388022987377, 2.90323204837276377919668223100, 4.41405114896647019014172284159, 6.24200839062188383211178157130, 8.250729638976887913016037153587, 9.184719958691400949498696138427, 10.58233346801730460193468039203, 12.70206417040598169769736553049, 13.39412574859986591689128795012, 14.57108042721280227095515991472, 15.63884988020789426083265049476

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