Properties

Degree 2
Conductor $ 2^{2} \cdot 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  − 322.·3-s + 3.12e3·5-s − 4.63e4·7-s − 7.32e4·9-s + 3.17e5·11-s + 1.96e6·13-s − 1.00e6·15-s + 6.70e6·17-s + 3.15e6·19-s + 1.49e7·21-s + 2.51e7·23-s + 9.76e6·25-s + 8.07e7·27-s + 1.66e8·29-s − 2.25e8·31-s − 1.02e8·33-s − 1.44e8·35-s − 4.84e7·37-s − 6.34e8·39-s − 5.76e8·41-s + 1.91e8·43-s − 2.28e8·45-s + 5.49e8·47-s + 1.68e8·49-s − 2.16e9·51-s + 1.16e9·53-s + 9.93e8·55-s + ⋯
L(s)  = 1  − 0.765·3-s + 0.447·5-s − 1.04·7-s − 0.413·9-s + 0.594·11-s + 1.47·13-s − 0.342·15-s + 1.14·17-s + 0.292·19-s + 0.797·21-s + 0.814·23-s + 0.199·25-s + 1.08·27-s + 1.50·29-s − 1.41·31-s − 0.455·33-s − 0.465·35-s − 0.114·37-s − 1.12·39-s − 0.776·41-s + 0.198·43-s − 0.184·45-s + 0.349·47-s + 0.0850·49-s − 0.876·51-s + 0.384·53-s + 0.266·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{20} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $1.38789$
$L(\frac12)$  $\approx$  $1.38789$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - 3.12e3T \)
good3 \( 1 + 322.T + 1.77e5T^{2} \)
7 \( 1 + 4.63e4T + 1.97e9T^{2} \)
11 \( 1 - 3.17e5T + 2.85e11T^{2} \)
13 \( 1 - 1.96e6T + 1.79e12T^{2} \)
17 \( 1 - 6.70e6T + 3.42e13T^{2} \)
19 \( 1 - 3.15e6T + 1.16e14T^{2} \)
23 \( 1 - 2.51e7T + 9.52e14T^{2} \)
29 \( 1 - 1.66e8T + 1.22e16T^{2} \)
31 \( 1 + 2.25e8T + 2.54e16T^{2} \)
37 \( 1 + 4.84e7T + 1.77e17T^{2} \)
41 \( 1 + 5.76e8T + 5.50e17T^{2} \)
43 \( 1 - 1.91e8T + 9.29e17T^{2} \)
47 \( 1 - 5.49e8T + 2.47e18T^{2} \)
53 \( 1 - 1.16e9T + 9.26e18T^{2} \)
59 \( 1 - 5.46e9T + 3.01e19T^{2} \)
61 \( 1 - 1.14e10T + 4.35e19T^{2} \)
67 \( 1 + 5.91e9T + 1.22e20T^{2} \)
71 \( 1 + 1.78e10T + 2.31e20T^{2} \)
73 \( 1 - 2.17e10T + 3.13e20T^{2} \)
79 \( 1 + 4.51e10T + 7.47e20T^{2} \)
83 \( 1 - 1.46e9T + 1.28e21T^{2} \)
89 \( 1 - 7.44e10T + 2.77e21T^{2} \)
97 \( 1 - 8.12e10T + 7.15e21T^{2} \)
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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

−16.03458385884730938452451840670, −14.26538754392787211991185845183, −12.95910073158939843814038225275, −11.64173366638452906321561310187, −10.30189389876211047645969176013, −8.846390771794035425364759206308, −6.62050569531529265957826017842, −5.58859370301800126600593293055, −3.34581210425957557176286113218, −0.931129079255647287645324693411, 0.931129079255647287645324693411, 3.34581210425957557176286113218, 5.58859370301800126600593293055, 6.62050569531529265957826017842, 8.846390771794035425364759206308, 10.30189389876211047645969176013, 11.64173366638452906321561310187, 12.95910073158939843814038225275, 14.26538754392787211991185845183, 16.03458385884730938452451840670

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