Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.0689 + 0.997i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 − 2.70i)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 + (17.3 + 14.4i)8-s + 13.2i·9-s + (−29.9 + 10.2i)10-s − 7.37i·11-s + (−5.93 + 29.0i)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  + (−0.287 − 0.957i)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 + (0.767 + 0.640i)8-s + 0.492i·9-s + (−0.945 + 0.324i)10-s − 0.202i·11-s + (−0.142 + 0.698i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0689 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0689 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.0689 + 0.997i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ 0.0689 + 0.997i)$
$L(2)$  $\approx$  $0.767720 - 0.716469i$
$L(\frac12)$  $\approx$  $0.767720 - 0.716469i$
$L(\frac{5}{2})$   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 \( 1 + (0.813 + 2.70i)T \)
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

−17.86881611474240504907332638523, −16.63249650174215171300059074201, −14.68511718395812670323069746590, −13.21251725643605531490933266890, −12.32776712472532127904171319281, −10.90226513816481719029646969690, −8.829687128124095342017662194647, −8.213996252559873179681797719489, −4.86748403966537872443889530926, −2.00358922228275972764457728887, 4.24966230655131450597625971537, 6.68156544128264921793062924906, 8.077323041626304831663517728510, 9.735239426817691807917541632119, 10.99740939227831827610610328226, 13.58156956375976491228886300974, 14.74449790639662437803013437598, 15.21369577111709536489444345271, 16.99465369633861029122404903318, 17.89191280119431106756791071822

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