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 − 140.·3-s + 256·4-s + 1.26e3·5-s + 2.24e3·6-s − 3.48e3·7-s − 4.09e3·8-s − 18.7·9-s − 2.02e4·10-s − 4.92e4·11-s − 3.58e4·12-s − 6.80e4·13-s + 5.57e4·14-s − 1.77e5·15-s + 6.55e4·16-s + 5.05e5·17-s + 299.·18-s + 1.30e5·19-s + 3.23e5·20-s + 4.89e5·21-s + 7.88e5·22-s − 5.85e5·23-s + 5.74e5·24-s − 3.55e5·25-s + 1.08e6·26-s + 2.76e6·27-s − 8.92e5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.999·3-s + 0.5·4-s + 0.904·5-s + 0.706·6-s − 0.548·7-s − 0.353·8-s − 0.000950·9-s − 0.639·10-s − 1.01·11-s − 0.499·12-s − 0.660·13-s + 0.388·14-s − 0.903·15-s + 0.250·16-s + 1.46·17-s + 0.000671·18-s + 0.229·19-s + 0.452·20-s + 0.548·21-s + 0.717·22-s − 0.435·23-s + 0.353·24-s − 0.182·25-s + 0.467·26-s + 1.00·27-s − 0.274·28-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

\(L(5)\) \(\approx\) \(0.813100\)
\(L(\frac12)\) \(\approx\) \(0.813100\)
\(L(\frac{11}{2})\) 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 \( 1 + 140.T + 1.96e4T^{2} \)
5 \( 1 - 1.26e3T + 1.95e6T^{2} \)
7 \( 1 + 3.48e3T + 4.03e7T^{2} \)
11 \( 1 + 4.92e4T + 2.35e9T^{2} \)
13 \( 1 + 6.80e4T + 1.06e10T^{2} \)
17 \( 1 - 5.05e5T + 1.18e11T^{2} \)
23 \( 1 + 5.85e5T + 1.80e12T^{2} \)
29 \( 1 - 2.62e6T + 1.45e13T^{2} \)
31 \( 1 - 3.53e6T + 2.64e13T^{2} \)
37 \( 1 - 1.85e7T + 1.29e14T^{2} \)
41 \( 1 - 2.18e7T + 3.27e14T^{2} \)
43 \( 1 + 1.12e7T + 5.02e14T^{2} \)
47 \( 1 - 1.54e7T + 1.11e15T^{2} \)
53 \( 1 - 7.05e6T + 3.29e15T^{2} \)
59 \( 1 - 2.90e7T + 8.66e15T^{2} \)
61 \( 1 - 1.44e8T + 1.16e16T^{2} \)
67 \( 1 - 2.65e7T + 2.72e16T^{2} \)
71 \( 1 + 4.27e7T + 4.58e16T^{2} \)
73 \( 1 - 3.24e8T + 5.88e16T^{2} \)
79 \( 1 + 8.88e7T + 1.19e17T^{2} \)
83 \( 1 - 6.29e7T + 1.86e17T^{2} \)
89 \( 1 + 4.54e7T + 3.50e17T^{2} \)
97 \( 1 - 1.64e9T + 7.60e17T^{2} \)
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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

−14.32143264703531427251772997057, −12.84469470676092631394726299823, −11.73323931206885318405757373078, −10.34570591117792494774103085855, −9.687027440152370595237789606900, −7.86442293455574043817575950135, −6.27182929119480261575562545553, −5.33803547761509390244254760403, −2.63360750445587434446796692078, −0.71196895381803246962145644293, 0.71196895381803246962145644293, 2.63360750445587434446796692078, 5.33803547761509390244254760403, 6.27182929119480261575562545553, 7.86442293455574043817575950135, 9.687027440152370595237789606900, 10.34570591117792494774103085855, 11.73323931206885318405757373078, 12.84469470676092631394726299823, 14.32143264703531427251772997057

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