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Properties

Label 2-38-19.18-c8-0-8
Degree $2$
Conductor $38$
Sign $-0.127 + 0.991i$
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 − 51.9i·3-s − 128.·4-s + 629.·5-s − 587.·6-s + 2.56e3·7-s + 1.44e3i·8-s + 3.86e3·9-s − 7.11e3i·10-s + 9.43e3·11-s + 6.65e3i·12-s − 7.15e3i·13-s − 2.90e4i·14-s − 3.26e4i·15-s + 1.63e4·16-s + 1.03e4·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.641i·3-s − 0.500·4-s + 1.00·5-s − 0.453·6-s + 1.06·7-s + 0.353i·8-s + 0.588·9-s − 0.711i·10-s + 0.644·11-s + 0.320i·12-s − 0.250i·13-s − 0.755i·14-s − 0.645i·15-s + 0.250·16-s + 0.123·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.127 + 0.991i$
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: \((2,\ 38,\ (\ :4),\ -0.127 + 0.991i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.60951 - 1.83049i\)
\(L(\frac12)\) \(\approx\) \(1.60951 - 1.83049i\)
\(L(5)\) 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 \( 1 + (1.29e5 + 1.66e4i)T \)
good3 \( 1 + 51.9iT - 6.56e3T^{2} \)
5 \( 1 - 629.T + 3.90e5T^{2} \)
7 \( 1 - 2.56e3T + 5.76e6T^{2} \)
11 \( 1 - 9.43e3T + 2.14e8T^{2} \)
13 \( 1 + 7.15e3iT - 8.15e8T^{2} \)
17 \( 1 - 1.03e4T + 6.97e9T^{2} \)
23 \( 1 - 8.40e4T + 7.83e10T^{2} \)
29 \( 1 + 6.68e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.42e5iT - 8.52e11T^{2} \)
37 \( 1 - 5.93e5iT - 3.51e12T^{2} \)
41 \( 1 - 3.30e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.37e6T + 1.16e13T^{2} \)
47 \( 1 + 8.88e5T + 2.38e13T^{2} \)
53 \( 1 - 6.29e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.98e7iT - 1.46e14T^{2} \)
61 \( 1 + 4.10e6T + 1.91e14T^{2} \)
67 \( 1 + 3.39e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.72e7iT - 6.45e14T^{2} \)
73 \( 1 + 6.59e4T + 8.06e14T^{2} \)
79 \( 1 - 3.84e6iT - 1.51e15T^{2} \)
83 \( 1 + 3.76e7T + 2.25e15T^{2} \)
89 \( 1 + 7.00e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.21e8iT - 7.83e15T^{2} \)
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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

−13.89113035620758083567094875689, −13.02706783942566095566583779464, −11.84702953917819827354580900823, −10.58094256237591662129893661223, −9.340653487533893933164219125974, −7.86450708806609093369163784064, −6.16103966074043553659523194924, −4.47168457954111691556024626723, −2.18414827441527387374887190954, −1.18064807316584360165746682965, 1.62299314660304728381226382043, 4.21216164705167057164748288988, 5.44323006276864889362145350405, 6.94664395791196841246770138003, 8.625734290913348218236099768024, 9.729416042343765668929878725381, 10.92231519649511291334087095335, 12.69691775430313999980431426449, 14.09368112811775613748344408797, 14.75964046719850333001769900943

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