Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.920 - 0.389i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 0.510i)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 + (−19.3 − 11.7i)8-s + 5.46i·9-s + (−31.6 + 0.899i)10-s − 47.0i·11-s + (18.6 + 41.5i)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  + (−0.983 − 0.180i)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 + (−0.855 − 0.517i)8-s + 0.202i·9-s + (−0.999 + 0.0284i)10-s − 1.28i·11-s + (0.449 + 1.00i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.920 - 0.389i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ 0.920 - 0.389i)$
$L(2)$  $\approx$  $0.922629 + 0.187307i$
$L(\frac12)$  $\approx$  $0.922629 + 0.187307i$
$L(\frac{5}{2})$   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 \( 1 + (2.78 + 0.510i)T \)
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

−18.06528834771601513016425541526, −16.57250002342352185544134088693, −15.73151580688921947150806196523, −14.25875957073140071857047132065, −12.55786558546095883013634867832, −10.64407147283354037402567126233, −9.315663151450989476924707477077, −8.761339985971014058958984736693, −6.21998371934022356453435387675, −2.89403133439151454233574161653, 2.15758103342252538819102954414, 6.52607985161147397187458486045, 7.68443594585726623506568097750, 9.419400006574715774991886483152, 10.44073548831785995159392234835, 12.70178199799571476384060143630, 13.87626873853453457571264852068, 15.23722280067712443264168892071, 16.95429441126145176392909871858, 17.77985279507936935910659355076

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