Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.890 + 0.455i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.70 − 0.813i)2-s + (−2.61 − 2.61i)3-s + (6.67 − 4.40i)4-s + (−0.435 + 11.1i)5-s + (−9.22 − 4.96i)6-s + (−17.7 + 17.7i)7-s + (14.4 − 17.3i)8-s − 13.2i·9-s + (7.91 + 30.6i)10-s − 7.37i·11-s + (−29.0 − 5.93i)12-s + (−2.68 + 2.68i)13-s + (−33.6 + 62.6i)14-s + (30.3 − 28.1i)15-s + (25.1 − 58.8i)16-s + (−20.2 − 20.2i)17-s + ⋯
L(s)  = 1  + (0.957 − 0.287i)2-s + (−0.503 − 0.503i)3-s + (0.834 − 0.551i)4-s + (−0.0389 + 0.999i)5-s + (−0.627 − 0.337i)6-s + (−0.959 + 0.959i)7-s + (0.640 − 0.767i)8-s − 0.492i·9-s + (0.250 + 0.968i)10-s − 0.202i·11-s + (−0.698 − 0.142i)12-s + (−0.0572 + 0.0572i)13-s + (−0.643 + 1.19i)14-s + (0.523 − 0.483i)15-s + (0.392 − 0.919i)16-s + (−0.288 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.890 + 0.455i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ 0.890 + 0.455i)$
$L(2)$  $\approx$  $1.41729 - 0.341235i$
$L(\frac12)$  $\approx$  $1.41729 - 0.341235i$
$L(\frac{5}{2})$   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 \( 1 + (-2.70 + 0.813i)T \)
5 \( 1 + (0.435 - 11.1i)T \)
good3 \( 1 + (2.61 + 2.61i)T + 27iT^{2} \)
7 \( 1 + (17.7 - 17.7i)T - 343iT^{2} \)
11 \( 1 + 7.37iT - 1.33e3T^{2} \)
13 \( 1 + (2.68 - 2.68i)T - 2.19e3iT^{2} \)
17 \( 1 + (20.2 + 20.2i)T + 4.91e3iT^{2} \)
19 \( 1 - 135.T + 6.85e3T^{2} \)
23 \( 1 + (71.0 + 71.0i)T + 1.21e4iT^{2} \)
29 \( 1 + 34.2iT - 2.43e4T^{2} \)
31 \( 1 - 187. iT - 2.97e4T^{2} \)
37 \( 1 + (-250. - 250. i)T + 5.06e4iT^{2} \)
41 \( 1 + 211.T + 6.89e4T^{2} \)
43 \( 1 + (-46.7 - 46.7i)T + 7.95e4iT^{2} \)
47 \( 1 + (-189. + 189. i)T - 1.03e5iT^{2} \)
53 \( 1 + (74.5 - 74.5i)T - 1.48e5iT^{2} \)
59 \( 1 - 101.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 + (34.7 - 34.7i)T - 3.00e5iT^{2} \)
71 \( 1 + 614. iT - 3.57e5T^{2} \)
73 \( 1 + (37.4 - 37.4i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (423. + 423. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (536. + 536. i)T + 9.12e5iT^{2} \)
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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

−18.15053734349966410797154992368, −16.11388004653613010191264512971, −15.09069164039564170599902342854, −13.73686238207739368742804246332, −12.33213665290293024797384334572, −11.51156679435831060116724522273, −9.826041882519395471945332774837, −6.90094636540833967842366730723, −5.87755051499836557867356258851, −3.06566795812459389820351851678, 4.10991926619355091410625798215, 5.61241536916767869930048852112, 7.56371739205221966535426462811, 9.886522970433804201978677728175, 11.47840074552586007040392685441, 12.91425202589678725531912227381, 13.80580975953522128164142216406, 15.77035175837558920781744992572, 16.37706990328811209678740952019, 17.28586556033312289397872614767

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