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Properties

Label 2-38-19.18-c8-0-2
Degree $2$
Conductor $38$
Sign $-0.211 - 0.977i$
Analytic cond. $15.4803$
Root an. cond. $3.93451$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.3i·2-s − 124. i·3-s − 128.·4-s − 419.·5-s + 1.40e3·6-s − 2.43e3·7-s − 1.44e3i·8-s − 8.87e3·9-s − 4.74e3i·10-s + 1.88e4·11-s + 1.59e4i·12-s + 5.21e4i·13-s − 2.75e4i·14-s + 5.20e4i·15-s + 1.63e4·16-s + 1.63e5·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.53i·3-s − 0.500·4-s − 0.670·5-s + 1.08·6-s − 1.01·7-s − 0.353i·8-s − 1.35·9-s − 0.474i·10-s + 1.28·11-s + 0.767i·12-s + 1.82i·13-s − 0.716i·14-s + 1.02i·15-s + 0.250·16-s + 1.95·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(15.4803\)
Root analytic conductor: \(3.93451\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :4),\ -0.211 - 0.977i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.430887 + 0.534336i\)
\(L(\frac12)\) \(\approx\) \(0.430887 + 0.534336i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 11.3iT \)
19 \( 1 + (1.27e5 - 2.76e4i)T \)
good3 \( 1 + 124. iT - 6.56e3T^{2} \)
5 \( 1 + 419.T + 3.90e5T^{2} \)
7 \( 1 + 2.43e3T + 5.76e6T^{2} \)
11 \( 1 - 1.88e4T + 2.14e8T^{2} \)
13 \( 1 - 5.21e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.63e5T + 6.97e9T^{2} \)
23 \( 1 + 2.94e5T + 7.83e10T^{2} \)
29 \( 1 - 1.01e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.88e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.31e6iT - 3.51e12T^{2} \)
41 \( 1 - 6.62e5iT - 7.98e12T^{2} \)
43 \( 1 + 1.24e6T + 1.16e13T^{2} \)
47 \( 1 + 3.35e6T + 2.38e13T^{2} \)
53 \( 1 - 6.52e6iT - 6.22e13T^{2} \)
59 \( 1 - 5.48e6iT - 1.46e14T^{2} \)
61 \( 1 - 8.15e6T + 1.91e14T^{2} \)
67 \( 1 + 1.27e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.99e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.59e7T + 8.06e14T^{2} \)
79 \( 1 - 5.24e7iT - 1.51e15T^{2} \)
83 \( 1 + 3.87e7T + 2.25e15T^{2} \)
89 \( 1 - 4.25e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.29e8iT - 7.83e15T^{2} \)
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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

−14.57905018066936300138587384508, −13.81419212060083418577479563756, −12.41766711098146600696770941754, −11.86459671046072586209902243376, −9.507915081872935726976350016671, −8.126268930414592689428545457962, −6.91411063179468248253642036646, −6.25339611276096634537787937240, −3.79350911026943278370632637462, −1.42739811022183394625809192111, 0.29408638140339584686293910147, 3.29703392081129302735568915080, 3.99458000593245559534363208685, 5.73945566210272342065576414682, 8.186270800670504716819701220588, 9.720058993105474262894476272327, 10.20677377125037290894963733265, 11.61051011713041638335841716435, 12.70784615970042271564071522383, 14.44535759848970874333612129050

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