Properties

Degree $2$
Conductor $38$
Sign $-1$
Motivic weight $7$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 3.23·3-s + 64·4-s − 312.·5-s + 25.9·6-s − 766.·7-s + 512·8-s − 2.17e3·9-s − 2.49e3·10-s + 252.·11-s + 207.·12-s − 1.06e3·13-s − 6.13e3·14-s − 1.01e3·15-s + 4.09e3·16-s − 1.87e4·17-s − 1.74e4·18-s − 6.85e3·19-s − 1.99e4·20-s − 2.48e3·21-s + 2.01e3·22-s + 1.15e4·23-s + 1.65e3·24-s + 1.95e4·25-s − 8.52e3·26-s − 1.41e4·27-s − 4.90e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.0692·3-s + 0.5·4-s − 1.11·5-s + 0.0489·6-s − 0.844·7-s + 0.353·8-s − 0.995·9-s − 0.790·10-s + 0.0570·11-s + 0.0346·12-s − 0.134·13-s − 0.597·14-s − 0.0774·15-s + 0.250·16-s − 0.926·17-s − 0.703·18-s − 0.229·19-s − 0.558·20-s − 0.0585·21-s + 0.0403·22-s + 0.198·23-s + 0.0244·24-s + 0.249·25-s − 0.0950·26-s − 0.138·27-s − 0.422·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-1$
Motivic weight: \(7\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
19 \( 1 + 6.85e3T \)
good3 \( 1 - 3.23T + 2.18e3T^{2} \)
5 \( 1 + 312.T + 7.81e4T^{2} \)
7 \( 1 + 766.T + 8.23e5T^{2} \)
11 \( 1 - 252.T + 1.94e7T^{2} \)
13 \( 1 + 1.06e3T + 6.27e7T^{2} \)
17 \( 1 + 1.87e4T + 4.10e8T^{2} \)
23 \( 1 - 1.15e4T + 3.40e9T^{2} \)
29 \( 1 + 4.62e4T + 1.72e10T^{2} \)
31 \( 1 - 4.68e4T + 2.75e10T^{2} \)
37 \( 1 + 1.82e5T + 9.49e10T^{2} \)
41 \( 1 - 8.19e5T + 1.94e11T^{2} \)
43 \( 1 - 4.77e5T + 2.71e11T^{2} \)
47 \( 1 - 9.92e5T + 5.06e11T^{2} \)
53 \( 1 + 8.52e5T + 1.17e12T^{2} \)
59 \( 1 + 1.92e6T + 2.48e12T^{2} \)
61 \( 1 - 2.09e5T + 3.14e12T^{2} \)
67 \( 1 + 2.32e6T + 6.06e12T^{2} \)
71 \( 1 + 5.37e6T + 9.09e12T^{2} \)
73 \( 1 + 3.71e6T + 1.10e13T^{2} \)
79 \( 1 + 1.30e6T + 1.92e13T^{2} \)
83 \( 1 + 6.51e6T + 2.71e13T^{2} \)
89 \( 1 - 3.99e6T + 4.42e13T^{2} \)
97 \( 1 + 6.90e6T + 8.07e13T^{2} \)
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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

−14.21830421122488668237391438244, −12.93745289358173204480144830356, −11.86790942981865475035807348446, −10.85855714937440245662825922801, −8.979835451662045915112157503590, −7.49298622933437872789889365905, −6.04600433220233326194543298214, −4.22970940818352579430758329214, −2.87694526840743978921992319847, 0, 2.87694526840743978921992319847, 4.22970940818352579430758329214, 6.04600433220233326194543298214, 7.49298622933437872789889365905, 8.979835451662045915112157503590, 10.85855714937440245662825922801, 11.86790942981865475035807348446, 12.93745289358173204480144830356, 14.21830421122488668237391438244

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