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  + 542.·3-s + 3.12e3·5-s + 4.96e4·7-s + 1.16e5·9-s − 1.94e4·11-s − 7.59e5·13-s + 1.69e6·15-s + 2.35e6·17-s + 1.62e7·19-s + 2.69e7·21-s + 3.07e7·23-s + 9.76e6·25-s − 3.26e7·27-s − 1.24e8·29-s + 2.87e8·31-s − 1.05e7·33-s + 1.55e8·35-s − 6.80e8·37-s − 4.11e8·39-s − 3.20e7·41-s − 1.32e9·43-s + 3.65e8·45-s − 2.20e9·47-s + 4.88e8·49-s + 1.27e9·51-s + 4.55e9·53-s − 6.07e7·55-s + ⋯
L(s)  = 1  + 1.28·3-s + 0.447·5-s + 1.11·7-s + 0.660·9-s − 0.0364·11-s − 0.567·13-s + 0.576·15-s + 0.402·17-s + 1.50·19-s + 1.43·21-s + 0.997·23-s + 0.199·25-s − 0.437·27-s − 1.12·29-s + 1.80·31-s − 0.0469·33-s + 0.499·35-s − 1.61·37-s − 0.731·39-s − 0.0432·41-s − 1.37·43-s + 0.295·45-s − 1.40·47-s + 0.247·49-s + 0.518·51-s + 1.49·53-s − 0.0162·55-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  =  $(2,\ 20,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $3.29744$
$L(\frac12)$  $\approx$  $3.29744$
$L(\frac{13}{2})$   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 \( 1 - 542.T + 1.77e5T^{2} \)
7 \( 1 - 4.96e4T + 1.97e9T^{2} \)
11 \( 1 + 1.94e4T + 2.85e11T^{2} \)
13 \( 1 + 7.59e5T + 1.79e12T^{2} \)
17 \( 1 - 2.35e6T + 3.42e13T^{2} \)
19 \( 1 - 1.62e7T + 1.16e14T^{2} \)
23 \( 1 - 3.07e7T + 9.52e14T^{2} \)
29 \( 1 + 1.24e8T + 1.22e16T^{2} \)
31 \( 1 - 2.87e8T + 2.54e16T^{2} \)
37 \( 1 + 6.80e8T + 1.77e17T^{2} \)
41 \( 1 + 3.20e7T + 5.50e17T^{2} \)
43 \( 1 + 1.32e9T + 9.29e17T^{2} \)
47 \( 1 + 2.20e9T + 2.47e18T^{2} \)
53 \( 1 - 4.55e9T + 9.26e18T^{2} \)
59 \( 1 + 1.70e9T + 3.01e19T^{2} \)
61 \( 1 + 6.50e9T + 4.35e19T^{2} \)
67 \( 1 - 6.87e7T + 1.22e20T^{2} \)
71 \( 1 + 2.55e10T + 2.31e20T^{2} \)
73 \( 1 - 1.19e10T + 3.13e20T^{2} \)
79 \( 1 + 9.68e9T + 7.47e20T^{2} \)
83 \( 1 + 1.07e10T + 1.28e21T^{2} \)
89 \( 1 + 7.08e10T + 2.77e21T^{2} \)
97 \( 1 + 1.53e10T + 7.15e21T^{2} \)
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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

−15.25411397526198273628201349383, −14.34170056291467338563775045040, −13.47597716219513099343278854800, −11.68490758417089141112057801834, −9.905910460617328377855063822446, −8.622021605888535278546947734776, −7.44814632292907492963114435548, −5.08692856252929801595846713850, −3.08193399406185628290849167281, −1.58922700445621024259624905057, 1.58922700445621024259624905057, 3.08193399406185628290849167281, 5.08692856252929801595846713850, 7.44814632292907492963114435548, 8.622021605888535278546947734776, 9.905910460617328377855063822446, 11.68490758417089141112057801834, 13.47597716219513099343278854800, 14.34170056291467338563775045040, 15.25411397526198273628201349383

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