Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.855 - 0.517i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.03 + 2.63i)2-s + (5.55 − 5.55i)3-s + (−5.87 + 5.43i)4-s + (−10.4 + 3.84i)5-s + (20.3 + 8.91i)6-s + (−1.14 − 1.14i)7-s + (−20.3 − 9.86i)8-s − 34.8i·9-s + (−20.9 − 23.6i)10-s − 27.0i·11-s + (−2.45 + 62.8i)12-s + (40.4 + 40.4i)13-s + (1.83 − 4.18i)14-s + (−37.0 + 79.7i)15-s + (5.00 − 63.8i)16-s + (−36.2 + 36.2i)17-s + ⋯
L(s)  = 1  + (0.364 + 0.931i)2-s + (1.06 − 1.06i)3-s + (−0.734 + 0.678i)4-s + (−0.939 + 0.343i)5-s + (1.38 + 0.606i)6-s + (−0.0616 − 0.0616i)7-s + (−0.899 − 0.436i)8-s − 1.28i·9-s + (−0.662 − 0.749i)10-s − 0.741i·11-s + (−0.0591 + 1.51i)12-s + (0.863 + 0.863i)13-s + (0.0349 − 0.0798i)14-s + (−0.637 + 1.37i)15-s + (0.0781 − 0.996i)16-s + (−0.517 + 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.855 - 0.517i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ 0.855 - 0.517i)$
$L(2)$  $\approx$  $1.33190 + 0.371138i$
$L(\frac12)$  $\approx$  $1.33190 + 0.371138i$
$L(\frac{5}{2})$   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 \( 1 + (-1.03 - 2.63i)T \)
5 \( 1 + (10.4 - 3.84i)T \)
good3 \( 1 + (-5.55 + 5.55i)T - 27iT^{2} \)
7 \( 1 + (1.14 + 1.14i)T + 343iT^{2} \)
11 \( 1 + 27.0iT - 1.33e3T^{2} \)
13 \( 1 + (-40.4 - 40.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (36.2 - 36.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 56.8T + 6.85e3T^{2} \)
23 \( 1 + (54.9 - 54.9i)T - 1.21e4iT^{2} \)
29 \( 1 - 57.1iT - 2.43e4T^{2} \)
31 \( 1 + 190. iT - 2.97e4T^{2} \)
37 \( 1 + (50.4 - 50.4i)T - 5.06e4iT^{2} \)
41 \( 1 + 71.5T + 6.89e4T^{2} \)
43 \( 1 + (66.9 - 66.9i)T - 7.95e4iT^{2} \)
47 \( 1 + (343. + 343. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-240. - 240. i)T + 1.48e5iT^{2} \)
59 \( 1 + 738.T + 2.05e5T^{2} \)
61 \( 1 + 187.T + 2.26e5T^{2} \)
67 \( 1 + (-576. - 576. i)T + 3.00e5iT^{2} \)
71 \( 1 + 157. iT - 3.57e5T^{2} \)
73 \( 1 + (-180. - 180. i)T + 3.89e5iT^{2} \)
79 \( 1 + 55.6T + 4.93e5T^{2} \)
83 \( 1 + (-858. + 858. i)T - 5.71e5iT^{2} \)
89 \( 1 + 158. iT - 7.04e5T^{2} \)
97 \( 1 + (1.11e3 - 1.11e3i)T - 9.12e5iT^{2} \)
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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

−18.24748149426735866095189137084, −16.42025702197973340121498561343, −15.21943901640500128088577515106, −14.04381626368414018450425463278, −13.23476583981247802535681824897, −11.71286980731045095899664950840, −8.790069629864098078359768572483, −7.82474583585897848069012564126, −6.57999626026445557564024249492, −3.59623023422900153389344886929, 3.30343824573709458557378305947, 4.66690016842414864037924154838, 8.323334158758097517523656759505, 9.532623484645942744216758494868, 10.86728320443560279539907719938, 12.42131574827743498282685984776, 13.88775020313447513222167681162, 15.16151145561317545770365985661, 15.89444940685217628237015663753, 18.12948571811663704418260965544

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