Properties

Label 2-38-19.18-c6-0-8
Degree $2$
Conductor $38$
Sign $-0.508 - 0.861i$
Analytic cond. $8.74205$
Root an. cond. $2.95669$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s − 13.8i·3-s − 32.0·4-s − 9.64·5-s − 78.3·6-s − 472.·7-s + 181. i·8-s + 537.·9-s + 54.5i·10-s − 1.63e3·11-s + 443. i·12-s + 2.50e3i·13-s + 2.67e3i·14-s + 133. i·15-s + 1.02e3·16-s + 1.38e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.513i·3-s − 0.500·4-s − 0.0771·5-s − 0.362·6-s − 1.37·7-s + 0.353i·8-s + 0.736·9-s + 0.0545i·10-s − 1.22·11-s + 0.256i·12-s + 1.14i·13-s + 0.974i·14-s + 0.0396i·15-s + 0.250·16-s + 0.282·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.508 - 0.861i$
Analytic conductor: \(8.74205\)
Root analytic conductor: \(2.95669\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3),\ -0.508 - 0.861i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0526066 + 0.0921067i\)
\(L(\frac12)\) \(\approx\) \(0.0526066 + 0.0921067i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
19 \( 1 + (5.90e3 - 3.48e3i)T \)
good3 \( 1 + 13.8iT - 729T^{2} \)
5 \( 1 + 9.64T + 1.56e4T^{2} \)
7 \( 1 + 472.T + 1.17e5T^{2} \)
11 \( 1 + 1.63e3T + 1.77e6T^{2} \)
13 \( 1 - 2.50e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.38e3T + 2.41e7T^{2} \)
23 \( 1 + 4.75e3T + 1.48e8T^{2} \)
29 \( 1 + 4.29e4iT - 5.94e8T^{2} \)
31 \( 1 - 7.23e3iT - 8.87e8T^{2} \)
37 \( 1 + 9.60e4iT - 2.56e9T^{2} \)
41 \( 1 + 2.72e3iT - 4.75e9T^{2} \)
43 \( 1 + 3.97e4T + 6.32e9T^{2} \)
47 \( 1 + 7.84e4T + 1.07e10T^{2} \)
53 \( 1 - 2.30e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.12e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.43e5T + 5.15e10T^{2} \)
67 \( 1 - 2.23e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.81e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.39e5T + 1.51e11T^{2} \)
79 \( 1 + 4.67e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.38e5T + 3.26e11T^{2} \)
89 \( 1 - 8.65e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.02e6iT - 8.32e11T^{2} \)
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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

−13.74177357495014286804170458814, −12.93687524240142153782897660265, −12.05509291035697622297353363176, −10.41553993573649166926916547749, −9.474305364344065934198794555980, −7.71264107826847770749853497724, −6.19023133993537112772124172249, −4.02479848563902741383354675949, −2.20031191778455532407894335979, −0.04982917952764402625666144158, 3.34342810804552054452169217127, 5.12486526502529185024049109391, 6.64386407791393237094142261135, 8.078152903437930763102126797724, 9.707517327203781170010945021123, 10.46354757572892532992475630982, 12.70999228735663540688160891657, 13.29578662207791866039911517938, 15.12886906024486088723825812990, 15.75546773064510961300323448073

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