Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.742 + 0.669i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s − 8i·4-s + (−2 + 11i)5-s + (−16 − 16i)8-s + 27i·9-s + (18 + 26i)10-s + (−37 − 37i)13-s − 64·16-s + (99 − 99i)17-s + (54 + 54i)18-s + (88 + 16i)20-s + (−117 − 44i)25-s − 148·26-s + 284i·29-s + (−128 + 128i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s i·4-s + (−0.178 + 0.983i)5-s + (−0.707 − 0.707i)8-s + i·9-s + (0.569 + 0.822i)10-s + (−0.789 − 0.789i)13-s − 16-s + (1.41 − 1.41i)17-s + (0.707 + 0.707i)18-s + (0.983 + 0.178i)20-s + (−0.936 − 0.351i)25-s − 1.11·26-s + 1.81i·29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.742 + 0.669i$
motivic weight  =  \(3\)
character  :  $\chi_{20} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :3/2),\ 0.742 + 0.669i)$
$L(2)$  $\approx$  $1.31007 - 0.503184i$
$L(\frac12)$  $\approx$  $1.31007 - 0.503184i$
$L(\frac{5}{2})$   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 \( 1 + (-2 + 2i)T \)
5 \( 1 + (2 - 11i)T \)
good3 \( 1 - 27iT^{2} \)
7 \( 1 + 343iT^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + (37 + 37i)T + 2.19e3iT^{2} \)
17 \( 1 + (-99 + 99i)T - 4.91e3iT^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 1.21e4iT^{2} \)
29 \( 1 - 284iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + (91 - 91i)T - 5.06e4iT^{2} \)
41 \( 1 - 472T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4iT^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 + (27 + 27i)T + 1.48e5iT^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 468T + 2.26e5T^{2} \)
67 \( 1 + 3.00e5iT^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-253 - 253i)T + 3.89e5iT^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5iT^{2} \)
89 \( 1 + 176iT - 7.04e5T^{2} \)
97 \( 1 + (611 - 611i)T - 9.12e5iT^{2} \)
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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

−18.14684558954369962380112005112, −16.17593689634869811986342933635, −14.74714739455204021924409479449, −13.88166295641359620815192458007, −12.34280805707255941107442787719, −10.99742868321278542442602360451, −9.946969841461214799388836065548, −7.36530271200470390972058886539, −5.26244609920122227016028362377, −2.92881682390944771765176714559, 4.12937926458756835599315736874, 5.95265495077236844226719802653, 7.83807535551204722256362042230, 9.345497442084030485072288587476, 11.94950548885687097205042938930, 12.69500147627342730624120166724, 14.27626125064912458928709606258, 15.38727699628562890512432007082, 16.69752237222435590939438327202, 17.45546087292726170577777845224

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