Properties

Label 2-38-1.1-c7-0-9
Degree $2$
Conductor $38$
Sign $-1$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 61.4·3-s + 64·4-s − 235.·5-s − 491.·6-s − 1.11e3·7-s − 512·8-s + 1.59e3·9-s + 1.88e3·10-s + 1.42e3·11-s + 3.93e3·12-s − 5.51e3·13-s + 8.89e3·14-s − 1.44e4·15-s + 4.09e3·16-s − 2.74e4·17-s − 1.27e4·18-s + 6.85e3·19-s − 1.50e4·20-s − 6.83e4·21-s − 1.14e4·22-s − 6.22e4·23-s − 3.14e4·24-s − 2.26e4·25-s + 4.41e4·26-s − 3.64e4·27-s − 7.11e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.31·3-s + 0.5·4-s − 0.842·5-s − 0.929·6-s − 1.22·7-s − 0.353·8-s + 0.729·9-s + 0.595·10-s + 0.323·11-s + 0.657·12-s − 0.696·13-s + 0.866·14-s − 1.10·15-s + 0.250·16-s − 1.35·17-s − 0.515·18-s + 0.229·19-s − 0.421·20-s − 1.61·21-s − 0.228·22-s − 1.06·23-s − 0.464·24-s − 0.290·25-s + 0.492·26-s − 0.356·27-s − 0.612·28-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: $-1$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
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 \( 1 - 61.4T + 2.18e3T^{2} \)
5 \( 1 + 235.T + 7.81e4T^{2} \)
7 \( 1 + 1.11e3T + 8.23e5T^{2} \)
11 \( 1 - 1.42e3T + 1.94e7T^{2} \)
13 \( 1 + 5.51e3T + 6.27e7T^{2} \)
17 \( 1 + 2.74e4T + 4.10e8T^{2} \)
23 \( 1 + 6.22e4T + 3.40e9T^{2} \)
29 \( 1 - 2.09e5T + 1.72e10T^{2} \)
31 \( 1 + 1.30e5T + 2.75e10T^{2} \)
37 \( 1 + 3.54e4T + 9.49e10T^{2} \)
41 \( 1 - 5.63e5T + 1.94e11T^{2} \)
43 \( 1 + 3.52e5T + 2.71e11T^{2} \)
47 \( 1 + 7.47e5T + 5.06e11T^{2} \)
53 \( 1 - 6.84e5T + 1.17e12T^{2} \)
59 \( 1 - 2.84e6T + 2.48e12T^{2} \)
61 \( 1 + 5.65e5T + 3.14e12T^{2} \)
67 \( 1 - 3.49e6T + 6.06e12T^{2} \)
71 \( 1 - 2.96e5T + 9.09e12T^{2} \)
73 \( 1 + 1.60e6T + 1.10e13T^{2} \)
79 \( 1 + 2.55e6T + 1.92e13T^{2} \)
83 \( 1 - 3.52e5T + 2.71e13T^{2} \)
89 \( 1 - 5.32e6T + 4.42e13T^{2} \)
97 \( 1 - 1.35e7T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.36230847015076929129336986001, −13.03653878327600035753842071550, −11.72232363991383297746253034670, −10.00156392296742469959240522939, −9.024774601216368293015170123697, −7.954296107405867270042512792760, −6.70690691654445538368764346621, −3.82034716797509714618683279097, −2.48349440097751772503419524990, 0, 2.48349440097751772503419524990, 3.82034716797509714618683279097, 6.70690691654445538368764346621, 7.954296107405867270042512792760, 9.024774601216368293015170123697, 10.00156392296742469959240522939, 11.72232363991383297746253034670, 13.03653878327600035753842071550, 14.36230847015076929129336986001

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