Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22·3-s − 25·5-s + 218·7-s + 241·9-s − 480·11-s − 622·13-s − 550·15-s + 186·17-s − 1.20e3·19-s + 4.79e3·21-s − 3.18e3·23-s + 625·25-s − 44·27-s + 5.52e3·29-s + 9.35e3·31-s − 1.05e4·33-s − 5.45e3·35-s + 5.61e3·37-s − 1.36e4·39-s − 1.43e4·41-s − 370·43-s − 6.02e3·45-s + 1.61e4·47-s + 3.07e4·49-s + 4.09e3·51-s − 4.37e3·53-s + 1.20e4·55-s + ⋯
L(s)  = 1  + 1.41·3-s − 0.447·5-s + 1.68·7-s + 0.991·9-s − 1.19·11-s − 1.02·13-s − 0.631·15-s + 0.156·17-s − 0.765·19-s + 2.37·21-s − 1.25·23-s + 1/5·25-s − 0.0116·27-s + 1.22·29-s + 1.74·31-s − 1.68·33-s − 0.752·35-s + 0.674·37-s − 1.44·39-s − 1.33·41-s − 0.0305·43-s − 0.443·45-s + 1.06·47-s + 1.82·49-s + 0.220·51-s − 0.213·53-s + 0.534·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{20} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $1.99658$
$L(\frac12)$  $\approx$  $1.99658$
$L(\frac{7}{2})$   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 \( 1 + p^{2} T \)
good3 \( 1 - 22 T + p^{5} T^{2} \)
7 \( 1 - 218 T + p^{5} T^{2} \)
11 \( 1 + 480 T + p^{5} T^{2} \)
13 \( 1 + 622 T + p^{5} T^{2} \)
17 \( 1 - 186 T + p^{5} T^{2} \)
19 \( 1 + 1204 T + p^{5} T^{2} \)
23 \( 1 + 3186 T + p^{5} T^{2} \)
29 \( 1 - 5526 T + p^{5} T^{2} \)
31 \( 1 - 9356 T + p^{5} T^{2} \)
37 \( 1 - 5618 T + p^{5} T^{2} \)
41 \( 1 + 14394 T + p^{5} T^{2} \)
43 \( 1 + 370 T + p^{5} T^{2} \)
47 \( 1 - 16146 T + p^{5} T^{2} \)
53 \( 1 + 4374 T + p^{5} T^{2} \)
59 \( 1 + 11748 T + p^{5} T^{2} \)
61 \( 1 - 13202 T + p^{5} T^{2} \)
67 \( 1 + 11542 T + p^{5} T^{2} \)
71 \( 1 + 29532 T + p^{5} T^{2} \)
73 \( 1 - 33698 T + p^{5} T^{2} \)
79 \( 1 - 31208 T + p^{5} T^{2} \)
83 \( 1 + 38466 T + p^{5} T^{2} \)
89 \( 1 - 119514 T + p^{5} T^{2} \)
97 \( 1 - 94658 T + p^{5} T^{2} \)
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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

−17.45760592998863266538651229557, −15.53456684155579359953972139624, −14.66844824519942955845592312212, −13.72210593724656417482917285761, −11.99534170829726721147213851992, −10.29024041268297841738990171808, −8.362876039403005985060153086809, −7.79164436867101309506219019452, −4.63666620757170252210011277593, −2.39577658256314220724904231661, 2.39577658256314220724904231661, 4.63666620757170252210011277593, 7.79164436867101309506219019452, 8.362876039403005985060153086809, 10.29024041268297841738990171808, 11.99534170829726721147213851992, 13.72210593724656417482917285761, 14.66844824519942955845592312212, 15.53456684155579359953972139624, 17.45760592998863266538651229557

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