Properties

Degree $2$
Conductor $38$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 17.4·3-s + 16·4-s − 79.4·5-s + 69.9·6-s + 132.·7-s − 64·8-s + 62.5·9-s + 317.·10-s + 311.·11-s − 279.·12-s + 901.·13-s − 531.·14-s + 1.38e3·15-s + 256·16-s − 157.·17-s − 250.·18-s + 361·19-s − 1.27e3·20-s − 2.32e3·21-s − 1.24e3·22-s − 2.52e3·23-s + 1.11e3·24-s + 3.18e3·25-s − 3.60e3·26-s + 3.15e3·27-s + 2.12e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.12·3-s + 0.5·4-s − 1.42·5-s + 0.792·6-s + 1.02·7-s − 0.353·8-s + 0.257·9-s + 1.00·10-s + 0.776·11-s − 0.560·12-s + 1.47·13-s − 0.724·14-s + 1.59·15-s + 0.250·16-s − 0.132·17-s − 0.182·18-s + 0.229·19-s − 0.710·20-s − 1.14·21-s − 0.548·22-s − 0.994·23-s + 0.396·24-s + 1.01·25-s − 1.04·26-s + 0.832·27-s + 0.512·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $1$
Motivic weight: \(5\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.657247\)
\(L(\frac12)\) \(\approx\) \(0.657247\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
19 \( 1 - 361T \)
good3 \( 1 + 17.4T + 243T^{2} \)
5 \( 1 + 79.4T + 3.12e3T^{2} \)
7 \( 1 - 132.T + 1.68e4T^{2} \)
11 \( 1 - 311.T + 1.61e5T^{2} \)
13 \( 1 - 901.T + 3.71e5T^{2} \)
17 \( 1 + 157.T + 1.41e6T^{2} \)
23 \( 1 + 2.52e3T + 6.43e6T^{2} \)
29 \( 1 - 4.73e3T + 2.05e7T^{2} \)
31 \( 1 + 6.58e3T + 2.86e7T^{2} \)
37 \( 1 - 8.50e3T + 6.93e7T^{2} \)
41 \( 1 - 1.97e4T + 1.15e8T^{2} \)
43 \( 1 - 1.09e4T + 1.47e8T^{2} \)
47 \( 1 - 1.50e4T + 2.29e8T^{2} \)
53 \( 1 - 2.16e4T + 4.18e8T^{2} \)
59 \( 1 + 4.06e4T + 7.14e8T^{2} \)
61 \( 1 - 6.15e3T + 8.44e8T^{2} \)
67 \( 1 - 6.27e4T + 1.35e9T^{2} \)
71 \( 1 + 5.53e4T + 1.80e9T^{2} \)
73 \( 1 + 4.85e4T + 2.07e9T^{2} \)
79 \( 1 - 3.10e4T + 3.07e9T^{2} \)
83 \( 1 - 4.10e4T + 3.93e9T^{2} \)
89 \( 1 + 1.70e4T + 5.58e9T^{2} \)
97 \( 1 - 1.39e5T + 8.58e9T^{2} \)
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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

−15.75426159982457036939000294038, −14.37729787115649742853691662155, −12.22143141025689843245357753400, −11.38162843749217152284541242011, −10.88283827358838766111517938128, −8.737522785476379508084403629336, −7.63645568361136861563448106403, −6.05181507622722288483448203211, −4.15004349959309532662649681447, −0.886440656638640508972368436622, 0.886440656638640508972368436622, 4.15004349959309532662649681447, 6.05181507622722288483448203211, 7.63645568361136861563448106403, 8.737522785476379508084403629336, 10.88283827358838766111517938128, 11.38162843749217152284541242011, 12.22143141025689843245357753400, 14.37729787115649742853691662155, 15.75426159982457036939000294038

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