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Properties

Label 2-38-19.18-c8-0-10
Degree $2$
Conductor $38$
Sign $-0.994 - 0.100i$
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 + 74.3i·3-s − 128.·4-s − 154.·5-s + 840.·6-s + 1.58e3·7-s + 1.44e3i·8-s + 1.03e3·9-s + 1.75e3i·10-s − 2.55e4·11-s − 9.51e3i·12-s − 1.03e4i·13-s − 1.79e4i·14-s − 1.15e4i·15-s + 1.63e4·16-s − 1.32e5·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.917i·3-s − 0.500·4-s − 0.247·5-s + 0.648·6-s + 0.660·7-s + 0.353i·8-s + 0.158·9-s + 0.175i·10-s − 1.74·11-s − 0.458i·12-s − 0.363i·13-s − 0.466i·14-s − 0.227i·15-s + 0.250·16-s − 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.994 - 0.100i$
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: \((2,\ 38,\ (\ :4),\ -0.994 - 0.100i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00140793 + 0.0278446i\)
\(L(\frac12)\) \(\approx\) \(0.00140793 + 0.0278446i\)
\(L(5)\) 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 \( 1 + (-1.31e4 + 1.29e5i)T \)
good3 \( 1 - 74.3iT - 6.56e3T^{2} \)
5 \( 1 + 154.T + 3.90e5T^{2} \)
7 \( 1 - 1.58e3T + 5.76e6T^{2} \)
11 \( 1 + 2.55e4T + 2.14e8T^{2} \)
13 \( 1 + 1.03e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.32e5T + 6.97e9T^{2} \)
23 \( 1 + 3.34e5T + 7.83e10T^{2} \)
29 \( 1 - 6.05e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.41e5iT - 8.52e11T^{2} \)
37 \( 1 + 2.45e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.38e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.53e6T + 1.16e13T^{2} \)
47 \( 1 - 5.01e6T + 2.38e13T^{2} \)
53 \( 1 - 1.18e7iT - 6.22e13T^{2} \)
59 \( 1 + 7.32e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.12e7T + 1.91e14T^{2} \)
67 \( 1 + 1.87e7iT - 4.06e14T^{2} \)
71 \( 1 - 2.17e7iT - 6.45e14T^{2} \)
73 \( 1 + 8.89e6T + 8.06e14T^{2} \)
79 \( 1 - 5.26e7iT - 1.51e15T^{2} \)
83 \( 1 - 3.76e7T + 2.25e15T^{2} \)
89 \( 1 - 1.06e8iT - 3.93e15T^{2} \)
97 \( 1 + 2.45e7iT - 7.83e15T^{2} \)
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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

−13.78278142868192014201991462028, −12.66447548852752119158182574305, −11.07157262868562272452286266695, −10.48740750749065009338306923581, −9.107707825687351930297552891746, −7.70882392148477589659429834414, −5.22103694578882533572449738822, −4.13863697752549449108337185977, −2.35160751052819013078876328531, −0.01041672211631904907691093810, 2.00496441871339651760443164403, 4.50970950348286816079192229791, 6.12899648117112428628130348924, 7.57916230738888114686447742825, 8.232067821104726553083702082416, 10.15714306062328064874792756613, 11.73721168313566563973454366111, 13.06361283524679331584607977062, 13.80351887138484681447300093701, 15.28863488206357135272482517246

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