Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.809 - 0.587i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 + 1.90i)2-s − 2.35i·3-s + (−3.23 + 2.35i)4-s − 2.23·5-s + (4.47 − 1.45i)6-s − 5.25i·7-s + (−6.47 − 4.70i)8-s + 3.47·9-s + (−1.38 − 4.25i)10-s + 19.9i·11-s + (5.52 + 7.60i)12-s − 8.47·13-s + (9.99 − 3.24i)14-s + 5.25i·15-s + (4.94 − 15.2i)16-s + 11.8·17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s − 0.783i·3-s + (−0.809 + 0.587i)4-s − 0.447·5-s + (0.745 − 0.242i)6-s − 0.751i·7-s + (−0.809 − 0.587i)8-s + 0.385·9-s + (−0.138 − 0.425i)10-s + 1.81i·11-s + (0.460 + 0.634i)12-s − 0.651·13-s + (0.714 − 0.232i)14-s + 0.350i·15-s + (0.309 − 0.951i)16-s + 0.699·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.809 - 0.587i$
motivic weight  =  \(2\)
character  :  $\chi_{20} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :1),\ 0.809 - 0.587i)$
$L(\frac{3}{2})$  $\approx$  $0.850569 + 0.276366i$
$L(\frac12)$  $\approx$  $0.850569 + 0.276366i$
$L(2)$   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 \( 1 + (-0.618 - 1.90i)T \)
5 \( 1 + 2.23T \)
good3 \( 1 + 2.35iT - 9T^{2} \)
7 \( 1 + 5.25iT - 49T^{2} \)
11 \( 1 - 19.9iT - 121T^{2} \)
13 \( 1 + 8.47T + 169T^{2} \)
17 \( 1 - 11.8T + 289T^{2} \)
19 \( 1 + 15.2iT - 361T^{2} \)
23 \( 1 + 0.555iT - 529T^{2} \)
29 \( 1 + 10.9T + 841T^{2} \)
31 \( 1 + 8.29iT - 961T^{2} \)
37 \( 1 + 18.3T + 1.36e3T^{2} \)
41 \( 1 + 14.5T + 1.68e3T^{2} \)
43 \( 1 - 22.2iT - 1.84e3T^{2} \)
47 \( 1 - 53.3iT - 2.20e3T^{2} \)
53 \( 1 + 66.3T + 2.80e3T^{2} \)
59 \( 1 - 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 90.1T + 3.72e3T^{2} \)
67 \( 1 - 50.2iT - 4.48e3T^{2} \)
71 \( 1 + 80.7iT - 5.04e3T^{2} \)
73 \( 1 + 5.55T + 5.32e3T^{2} \)
79 \( 1 + 13.8iT - 6.24e3T^{2} \)
83 \( 1 + 76.2iT - 6.88e3T^{2} \)
89 \( 1 + 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8T + 9.40e3T^{2} \)
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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

−17.96639225779597393254419667225, −17.10977597074751866119696618412, −15.62779135732896705591601604435, −14.51309729259349045552606087342, −13.08663694101274076841807556316, −12.21303170706440393411827475958, −9.788602370359588837916928515174, −7.64047041750971698148875835846, −6.98214116978119855428849482800, −4.53806032610858065508113104593, 3.53233895242241329213728963533, 5.43441430703993799926017740945, 8.592821475085787209603572063168, 10.01272837542119754235070824034, 11.29415588817075777185017400195, 12.49875361328906704754609653084, 14.11965764076519389585473530947, 15.34922760887369189913514808858, 16.60144260466840259826373441057, 18.58585532578778796125086203024

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