Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.7·2-s − 59.7·3-s + 222.·4-s − 125·5-s − 1.11e3·6-s + 438.·7-s + 1.76e3·8-s + 1.38e3·9-s − 2.33e3·10-s + 5.75e3·11-s − 1.32e4·12-s − 3.53e3·13-s + 8.20e3·14-s + 7.46e3·15-s + 4.59e3·16-s − 2.39e4·17-s + 2.58e4·18-s + 1.65e4·19-s − 2.77e4·20-s − 2.61e4·21-s + 1.07e5·22-s − 6.56e4·23-s − 1.05e5·24-s + 1.56e4·25-s − 6.60e4·26-s + 4.80e4·27-s + 9.74e4·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 1.27·3-s + 1.73·4-s − 0.447·5-s − 2.11·6-s + 0.482·7-s + 1.21·8-s + 0.631·9-s − 0.739·10-s + 1.30·11-s − 2.21·12-s − 0.445·13-s + 0.798·14-s + 0.571·15-s + 0.280·16-s − 1.18·17-s + 1.04·18-s + 0.554·19-s − 0.776·20-s − 0.616·21-s + 2.15·22-s − 1.12·23-s − 1.55·24-s + 0.199·25-s − 0.737·26-s + 0.470·27-s + 0.838·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $1.78069$
$L(\frac12)$  $\approx$  $1.78069$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + 125T \)
good2 \( 1 - 18.7T + 128T^{2} \)
3 \( 1 + 59.7T + 2.18e3T^{2} \)
7 \( 1 - 438.T + 8.23e5T^{2} \)
11 \( 1 - 5.75e3T + 1.94e7T^{2} \)
13 \( 1 + 3.53e3T + 6.27e7T^{2} \)
17 \( 1 + 2.39e4T + 4.10e8T^{2} \)
19 \( 1 - 1.65e4T + 8.93e8T^{2} \)
23 \( 1 + 6.56e4T + 3.40e9T^{2} \)
29 \( 1 - 1.34e5T + 1.72e10T^{2} \)
31 \( 1 - 1.29e5T + 2.75e10T^{2} \)
37 \( 1 - 1.61e5T + 9.49e10T^{2} \)
41 \( 1 + 3.62e5T + 1.94e11T^{2} \)
43 \( 1 - 5.88e5T + 2.71e11T^{2} \)
47 \( 1 - 3.43e5T + 5.06e11T^{2} \)
53 \( 1 + 1.66e6T + 1.17e12T^{2} \)
59 \( 1 + 2.54e6T + 2.48e12T^{2} \)
61 \( 1 - 2.52e6T + 3.14e12T^{2} \)
67 \( 1 - 1.56e6T + 6.06e12T^{2} \)
71 \( 1 + 2.99e5T + 9.09e12T^{2} \)
73 \( 1 - 3.12e5T + 1.10e13T^{2} \)
79 \( 1 + 1.95e6T + 1.92e13T^{2} \)
83 \( 1 + 6.21e5T + 2.71e13T^{2} \)
89 \( 1 - 5.78e6T + 4.42e13T^{2} \)
97 \( 1 - 7.20e6T + 8.07e13T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−22.54060403502313847767575986943, −21.90639216795998310222946270991, −20.05309366556374309499376322086, −17.49591101229876690110058287826, −15.85325839574730429144282105618, −14.21000976480262357162330942802, −12.19035669906091329910107758905, −11.34895122175493616646545967309, −6.41028214690635329556348984161, −4.56624038341865425167686208504, 4.56624038341865425167686208504, 6.41028214690635329556348984161, 11.34895122175493616646545967309, 12.19035669906091329910107758905, 14.21000976480262357162330942802, 15.85325839574730429144282105618, 17.49591101229876690110058287826, 20.05309366556374309499376322086, 21.90639216795998310222946270991, 22.54060403502313847767575986943

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