Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 63.7·2-s + 283.·3-s + 2.01e3·4-s − 3.12e3·5-s + 1.80e4·6-s + 4.19e4·7-s − 2.20e3·8-s − 9.69e4·9-s − 1.99e5·10-s − 9.57e5·11-s + 5.70e5·12-s + 1.39e6·13-s + 2.67e6·14-s − 8.85e5·15-s − 4.26e6·16-s + 3.76e6·17-s − 6.17e6·18-s + 9.41e6·19-s − 6.29e6·20-s + 1.18e7·21-s − 6.10e7·22-s + 3.02e7·23-s − 6.24e5·24-s + 9.76e6·25-s + 8.86e7·26-s − 7.76e7·27-s + 8.44e7·28-s + ⋯
L(s)  = 1  + 1.40·2-s + 0.672·3-s + 0.983·4-s − 0.447·5-s + 0.947·6-s + 0.942·7-s − 0.0237·8-s − 0.547·9-s − 0.629·10-s − 1.79·11-s + 0.661·12-s + 1.03·13-s + 1.32·14-s − 0.300·15-s − 1.01·16-s + 0.643·17-s − 0.770·18-s + 0.871·19-s − 0.439·20-s + 0.634·21-s − 2.52·22-s + 0.981·23-s − 0.0160·24-s + 0.199·25-s + 1.46·26-s − 1.04·27-s + 0.926·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $3.03157$
$L(\frac12)$  $\approx$  $3.03157$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + 3.12e3T \)
good2 \( 1 - 63.7T + 2.04e3T^{2} \)
3 \( 1 - 283.T + 1.77e5T^{2} \)
7 \( 1 - 4.19e4T + 1.97e9T^{2} \)
11 \( 1 + 9.57e5T + 2.85e11T^{2} \)
13 \( 1 - 1.39e6T + 1.79e12T^{2} \)
17 \( 1 - 3.76e6T + 3.42e13T^{2} \)
19 \( 1 - 9.41e6T + 1.16e14T^{2} \)
23 \( 1 - 3.02e7T + 9.52e14T^{2} \)
29 \( 1 - 1.03e8T + 1.22e16T^{2} \)
31 \( 1 + 5.48e7T + 2.54e16T^{2} \)
37 \( 1 - 4.78e8T + 1.77e17T^{2} \)
41 \( 1 + 9.29e8T + 5.50e17T^{2} \)
43 \( 1 + 2.68e7T + 9.29e17T^{2} \)
47 \( 1 + 1.20e9T + 2.47e18T^{2} \)
53 \( 1 + 4.02e9T + 9.26e18T^{2} \)
59 \( 1 - 7.97e9T + 3.01e19T^{2} \)
61 \( 1 + 2.07e9T + 4.35e19T^{2} \)
67 \( 1 - 5.61e9T + 1.22e20T^{2} \)
71 \( 1 - 1.51e10T + 2.31e20T^{2} \)
73 \( 1 + 6.64e9T + 3.13e20T^{2} \)
79 \( 1 + 1.57e10T + 7.47e20T^{2} \)
83 \( 1 + 2.04e10T + 1.28e21T^{2} \)
89 \( 1 + 4.21e10T + 2.77e21T^{2} \)
97 \( 1 - 1.10e11T + 7.15e21T^{2} \)
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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

−21.18229820374676738356905064732, −20.48001615834835926319206611091, −18.33336979256135259792305720362, −15.69645877528885658226412516390, −14.45058408175387493894649660214, −13.21307843486803838558402299942, −11.33877513350072406211360404469, −8.145927575509375195319117425658, −5.20084135453372950822890389112, −3.07033621261827739461875958973, 3.07033621261827739461875958973, 5.20084135453372950822890389112, 8.145927575509375195319117425658, 11.33877513350072406211360404469, 13.21307843486803838558402299942, 14.45058408175387493894649660214, 15.69645877528885658226412516390, 18.33336979256135259792305720362, 20.48001615834835926319206611091, 21.18229820374676738356905064732

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