Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 2·3-s + 16·4-s + 25·5-s − 8·6-s − 82·7-s + 64·8-s − 77·9-s + 100·10-s − 32·12-s − 328·14-s − 50·15-s + 256·16-s − 308·18-s + 400·20-s + 164·21-s + 878·23-s − 128·24-s + 625·25-s + 316·27-s − 1.31e3·28-s − 1.19e3·29-s − 200·30-s + 1.02e3·32-s − 2.05e3·35-s − 1.23e3·36-s + 1.60e3·40-s + ⋯
L(s)  = 1  + 2-s − 2/9·3-s + 4-s + 5-s − 2/9·6-s − 1.67·7-s + 8-s − 0.950·9-s + 10-s − 2/9·12-s − 1.67·14-s − 2/9·15-s + 16-s − 0.950·18-s + 20-s + 0.371·21-s + 1.65·23-s − 2/9·24-s + 25-s + 0.433·27-s − 1.67·28-s − 1.42·29-s − 2/9·30-s + 32-s − 1.67·35-s − 0.950·36-s + 40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  $\chi_{20} (19, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :2),\ 1)$
$L(\frac{5}{2})$  $\approx$  $1.97006$
$L(\frac12)$  $\approx$  $1.97006$
$L(3)$   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 \( 1 - p^{2} T \)
5 \( 1 - p^{2} T \)
good3 \( 1 + 2 T + p^{4} T^{2} \)
7 \( 1 + 82 T + p^{4} T^{2} \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( 1 - 878 T + p^{4} T^{2} \)
29 \( 1 + 1198 T + p^{4} T^{2} \)
31 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( 1 - 482 T + p^{4} T^{2} \)
43 \( 1 - 2078 T + p^{4} T^{2} \)
47 \( 1 + 4402 T + p^{4} T^{2} \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 + 4078 T + p^{4} T^{2} \)
67 \( 1 - 4478 T + p^{4} T^{2} \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
79 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
83 \( 1 + 8002 T + p^{4} T^{2} \)
89 \( 1 - 4322 T + p^{4} T^{2} \)
97 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
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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

−17.14945125156150549299448115698, −16.30610719695318278337473083959, −14.79053355480368634301448882234, −13.46412531471445479079452463682, −12.65963603095665935286461435269, −10.95685224310127200665425694761, −9.435043939049754964564599934274, −6.68865815789613896621709126570, −5.58233937358503204092187061365, −2.99793634881491928018432774592, 2.99793634881491928018432774592, 5.58233937358503204092187061365, 6.68865815789613896621709126570, 9.435043939049754964564599934274, 10.95685224310127200665425694761, 12.65963603095665935286461435269, 13.46412531471445479079452463682, 14.79053355480368634301448882234, 16.30610719695318278337473083959, 17.14945125156150549299448115698

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