Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $-0.429 - 0.902i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.510 + 2.78i)2-s + (−4.02 + 4.02i)3-s + (−7.47 − 2.83i)4-s + (10.9 + 2.32i)5-s + (−9.15 − 13.2i)6-s + (14.4 + 14.4i)7-s + (11.7 − 19.3i)8-s − 5.46i·9-s + (−12.0 + 29.2i)10-s − 47.0i·11-s + (41.5 − 18.6i)12-s + (−8.79 − 8.79i)13-s + (−47.5 + 32.8i)14-s + (−53.4 + 34.6i)15-s + (47.8 + 42.4i)16-s + (−26.4 + 26.4i)17-s + ⋯
L(s)  = 1  + (−0.180 + 0.983i)2-s + (−0.775 + 0.775i)3-s + (−0.934 − 0.354i)4-s + (0.978 + 0.208i)5-s + (−0.622 − 0.902i)6-s + (0.779 + 0.779i)7-s + (0.517 − 0.855i)8-s − 0.202i·9-s + (−0.381 + 0.924i)10-s − 1.28i·11-s + (1.00 − 0.449i)12-s + (−0.187 − 0.187i)13-s + (−0.907 + 0.626i)14-s + (−0.919 + 0.596i)15-s + (0.747 + 0.663i)16-s + (−0.377 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.429 - 0.902i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ -0.429 - 0.902i)$
$L(2)$  $\approx$  $0.466631 + 0.739033i$
$L(\frac12)$  $\approx$  $0.466631 + 0.739033i$
$L(\frac{5}{2})$   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 \( 1 + (0.510 - 2.78i)T \)
5 \( 1 + (-10.9 - 2.32i)T \)
good3 \( 1 + (4.02 - 4.02i)T - 27iT^{2} \)
7 \( 1 + (-14.4 - 14.4i)T + 343iT^{2} \)
11 \( 1 + 47.0iT - 1.33e3T^{2} \)
13 \( 1 + (8.79 + 8.79i)T + 2.19e3iT^{2} \)
17 \( 1 + (26.4 - 26.4i)T - 4.91e3iT^{2} \)
19 \( 1 - 49.8T + 6.85e3T^{2} \)
23 \( 1 + (-41.2 + 41.2i)T - 1.21e4iT^{2} \)
29 \( 1 + 247. iT - 2.43e4T^{2} \)
31 \( 1 - 62.3iT - 2.97e4T^{2} \)
37 \( 1 + (73.2 - 73.2i)T - 5.06e4iT^{2} \)
41 \( 1 - 118.T + 6.89e4T^{2} \)
43 \( 1 + (245. - 245. i)T - 7.95e4iT^{2} \)
47 \( 1 + (125. + 125. i)T + 1.03e5iT^{2} \)
53 \( 1 + (326. + 326. i)T + 1.48e5iT^{2} \)
59 \( 1 - 365.T + 2.05e5T^{2} \)
61 \( 1 + 268.T + 2.26e5T^{2} \)
67 \( 1 + (-112. - 112. i)T + 3.00e5iT^{2} \)
71 \( 1 - 559. iT - 3.57e5T^{2} \)
73 \( 1 + (-215. - 215. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + (-592. + 592. i)T - 5.71e5iT^{2} \)
89 \( 1 - 552. iT - 7.04e5T^{2} \)
97 \( 1 + (-460. + 460. i)T - 9.12e5iT^{2} \)
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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

−17.82774945949674882394337286596, −16.97164713760731699950011859544, −15.89353724487323927610275982809, −14.67404770927266189144315739987, −13.43960017639732019534965628059, −11.24698935806875743926966083811, −9.915470233499977202168121370849, −8.439904173195141790459013438083, −6.08003447560574532295963258695, −5.09659175642776025805495108126, 1.51891437296429275105910588701, 4.98389133627466619250463836693, 7.23190358823218872578164335035, 9.399528730580484261776589120337, 10.78819775661994631196785931301, 12.10321468443508349104157916001, 13.14562394125720404168773497423, 14.34179980883879406150067092396, 17.01398480099287287596753840720, 17.69228910626314187536542704257

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