Properties

Degree $2$
Conductor $38$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s − 66.6·3-s + 256·4-s − 2.41e3·5-s + 1.06e3·6-s − 1.53e3·7-s − 4.09e3·8-s − 1.52e4·9-s + 3.86e4·10-s − 5.81e3·11-s − 1.70e4·12-s − 1.35e5·13-s + 2.45e4·14-s + 1.61e5·15-s + 6.55e4·16-s − 4.48e5·17-s + 2.43e5·18-s + 1.30e5·19-s − 6.19e5·20-s + 1.02e5·21-s + 9.30e4·22-s + 2.07e6·23-s + 2.72e5·24-s + 3.89e6·25-s + 2.16e6·26-s + 2.32e6·27-s − 3.92e5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.474·3-s + 0.5·4-s − 1.73·5-s + 0.335·6-s − 0.241·7-s − 0.353·8-s − 0.774·9-s + 1.22·10-s − 0.119·11-s − 0.237·12-s − 1.31·13-s + 0.170·14-s + 0.821·15-s + 0.250·16-s − 1.30·17-s + 0.547·18-s + 0.229·19-s − 0.865·20-s + 0.114·21-s + 0.0846·22-s + 1.54·23-s + 0.167·24-s + 1.99·25-s + 0.930·26-s + 0.842·27-s − 0.120·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $1$
Motivic weight: \(9\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.270241\)
\(L(\frac12)\) \(\approx\) \(0.270241\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16T \)
19 \( 1 - 1.30e5T \)
good3 \( 1 + 66.6T + 1.96e4T^{2} \)
5 \( 1 + 2.41e3T + 1.95e6T^{2} \)
7 \( 1 + 1.53e3T + 4.03e7T^{2} \)
11 \( 1 + 5.81e3T + 2.35e9T^{2} \)
13 \( 1 + 1.35e5T + 1.06e10T^{2} \)
17 \( 1 + 4.48e5T + 1.18e11T^{2} \)
23 \( 1 - 2.07e6T + 1.80e12T^{2} \)
29 \( 1 + 6.43e5T + 1.45e13T^{2} \)
31 \( 1 + 1.94e5T + 2.64e13T^{2} \)
37 \( 1 - 9.46e6T + 1.29e14T^{2} \)
41 \( 1 + 2.34e7T + 3.27e14T^{2} \)
43 \( 1 - 3.80e7T + 5.02e14T^{2} \)
47 \( 1 + 2.28e7T + 1.11e15T^{2} \)
53 \( 1 + 6.65e7T + 3.29e15T^{2} \)
59 \( 1 + 9.14e7T + 8.66e15T^{2} \)
61 \( 1 + 4.02e7T + 1.16e16T^{2} \)
67 \( 1 - 2.42e8T + 2.72e16T^{2} \)
71 \( 1 + 1.83e8T + 4.58e16T^{2} \)
73 \( 1 + 1.39e8T + 5.88e16T^{2} \)
79 \( 1 - 2.51e8T + 1.19e17T^{2} \)
83 \( 1 + 1.30e8T + 1.86e17T^{2} \)
89 \( 1 - 7.10e8T + 3.50e17T^{2} \)
97 \( 1 + 1.31e9T + 7.60e17T^{2} \)
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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

−14.74651104797576572081853058812, −12.64613952086262162101527115087, −11.58682098191906831976607209184, −10.91131240167889948648079635276, −9.129828211633625094292699040503, −7.893294801999888533030083100752, −6.80372721054149361834684572273, −4.77690315618524440666991753821, −2.95073188165454631225153947198, −0.38491129285373576536360182428, 0.38491129285373576536360182428, 2.95073188165454631225153947198, 4.77690315618524440666991753821, 6.80372721054149361834684572273, 7.893294801999888533030083100752, 9.129828211633625094292699040503, 10.91131240167889948648079635276, 11.58682098191906831976607209184, 12.64613952086262162101527115087, 14.74651104797576572081853058812

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