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Properties

Label 2-38-19.18-c6-0-9
Degree $2$
Conductor $38$
Sign $-0.480 - 0.877i$
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 − 44.5i·3-s − 32.0·4-s − 145.·5-s − 252.·6-s + 126.·7-s + 181. i·8-s − 1.25e3·9-s + 820. i·10-s + 2.26e3·11-s + 1.42e3i·12-s − 1.37e3i·13-s − 716. i·14-s + 6.46e3i·15-s + 1.02e3·16-s − 7.18e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.65i·3-s − 0.500·4-s − 1.16·5-s − 1.16·6-s + 0.369·7-s + 0.353i·8-s − 1.72·9-s + 0.820i·10-s + 1.70·11-s + 0.825i·12-s − 0.627i·13-s − 0.260i·14-s + 1.91i·15-s + 0.250·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.480 - 0.877i$
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: \((2,\ 38,\ (\ :3),\ -0.480 - 0.877i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.388559 + 0.655829i\)
\(L(\frac12)\) \(\approx\) \(0.388559 + 0.655829i\)
\(L(4)\) 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 \( 1 + (6.01e3 - 3.29e3i)T \)
good3 \( 1 + 44.5iT - 729T^{2} \)
5 \( 1 + 145.T + 1.56e4T^{2} \)
7 \( 1 - 126.T + 1.17e5T^{2} \)
11 \( 1 - 2.26e3T + 1.77e6T^{2} \)
13 \( 1 + 1.37e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.18e3T + 2.41e7T^{2} \)
23 \( 1 + 7.10e3T + 1.48e8T^{2} \)
29 \( 1 - 2.29e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.98e4iT - 8.87e8T^{2} \)
37 \( 1 + 5.82e3iT - 2.56e9T^{2} \)
41 \( 1 + 5.46e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.20e4T + 6.32e9T^{2} \)
47 \( 1 - 1.55e5T + 1.07e10T^{2} \)
53 \( 1 + 1.22e5iT - 2.21e10T^{2} \)
59 \( 1 - 5.42e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.54e5T + 5.15e10T^{2} \)
67 \( 1 + 5.54e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.41e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.83e5T + 1.51e11T^{2} \)
79 \( 1 + 5.08e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.01e5T + 3.26e11T^{2} \)
89 \( 1 + 3.63e5iT - 4.96e11T^{2} \)
97 \( 1 - 4.95e5iT - 8.32e11T^{2} \)
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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

−13.92860126352461408337162229495, −12.71828076127366756807003510525, −11.91073752736752005869843937017, −11.12744234154961376212247755255, −8.810212555520926000872960767866, −7.76335882006796373785608344004, −6.43954044108659574816563149680, −3.98386972499032034080511277307, −1.90564153685479980898852652225, −0.38400517740631515784228467933, 3.96878846235512310681459364185, 4.52800248332534957922935798640, 6.60915539970139932250554727264, 8.496803180468473510531297944312, 9.317693409417671317591757240976, 10.93391645581042842517954817885, 11.86580449339070782933896586947, 14.07197694563250854964843829546, 15.04387934897007874207231282177, 15.67571717085903336498631756535

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