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Properties

Label 2-38-19.18-c8-0-6
Degree $2$
Conductor $38$
Sign $0.952 + 0.304i$
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 + 10.7i·3-s − 128.·4-s − 919.·5-s − 121.·6-s − 343.·7-s − 1.44e3i·8-s + 6.44e3·9-s − 1.04e4i·10-s + 4.19e3·11-s − 1.37e3i·12-s − 1.65e4i·13-s − 3.88e3i·14-s − 9.88e3i·15-s + 1.63e4·16-s + 5.02e3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.132i·3-s − 0.500·4-s − 1.47·5-s − 0.0938·6-s − 0.143·7-s − 0.353i·8-s + 0.982·9-s − 1.04i·10-s + 0.286·11-s − 0.0663i·12-s − 0.577i·13-s − 0.101i·14-s − 0.195i·15-s + 0.250·16-s + 0.0601·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.952 + 0.304i$
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: \((2,\ 38,\ (\ :4),\ 0.952 + 0.304i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.09412 - 0.170925i\)
\(L(\frac12)\) \(\approx\) \(1.09412 - 0.170925i\)
\(L(5)\) 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 \( 1 + (-3.97e4 + 1.24e5i)T \)
good3 \( 1 - 10.7iT - 6.56e3T^{2} \)
5 \( 1 + 919.T + 3.90e5T^{2} \)
7 \( 1 + 343.T + 5.76e6T^{2} \)
11 \( 1 - 4.19e3T + 2.14e8T^{2} \)
13 \( 1 + 1.65e4iT - 8.15e8T^{2} \)
17 \( 1 - 5.02e3T + 6.97e9T^{2} \)
23 \( 1 - 3.50e5T + 7.83e10T^{2} \)
29 \( 1 + 4.84e5iT - 5.00e11T^{2} \)
31 \( 1 - 2.63e5iT - 8.52e11T^{2} \)
37 \( 1 + 2.12e6iT - 3.51e12T^{2} \)
41 \( 1 + 3.03e6iT - 7.98e12T^{2} \)
43 \( 1 + 2.35e6T + 1.16e13T^{2} \)
47 \( 1 - 7.27e5T + 2.38e13T^{2} \)
53 \( 1 + 6.39e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.53e7iT - 1.46e14T^{2} \)
61 \( 1 - 9.63e5T + 1.91e14T^{2} \)
67 \( 1 + 3.33e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.44e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.36e7T + 8.06e14T^{2} \)
79 \( 1 + 5.45e7iT - 1.51e15T^{2} \)
83 \( 1 - 4.12e7T + 2.25e15T^{2} \)
89 \( 1 - 7.15e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.33e8iT - 7.83e15T^{2} \)
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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

−14.96859178946711461189154494754, −13.30068312491287019267964197492, −12.17831292147213026232644413706, −10.80864606573554558119617033618, −9.203206199231065592249705909032, −7.81381364671335558894335836916, −6.89629710023207023855073860273, −4.85625592345526228008546789916, −3.59190402312199834745690937892, −0.54029770136152289601129596444, 1.22087993071193455536783916342, 3.42928961581928940902839229002, 4.57733079061530582742146886350, 6.97968284984224158416883491922, 8.244810488887732883456988814550, 9.746893415519474781502972707390, 11.17016487421458080739624980864, 12.07929928004667369324506048854, 13.04914114612324881124972693208, 14.60266216447864710954304647272

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