Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.995 + 0.0898i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)3-s + (−3 + 4i)5-s + (−7 − 7i)7-s + 7i·9-s + 10·11-s + (9 − 9i)13-s + (1 + 7i)15-s + (1 + i)17-s − 8i·19-s − 14·21-s + (−23 + 23i)23-s + (−7 − 24i)25-s + (16 + 16i)27-s − 8i·29-s − 14·31-s + ⋯
L(s)  = 1  + (0.333 − 0.333i)3-s + (−0.600 + 0.800i)5-s + (−1 − i)7-s + 0.777i·9-s + 0.909·11-s + (0.692 − 0.692i)13-s + (0.0666 + 0.466i)15-s + (0.0588 + 0.0588i)17-s − 0.421i·19-s − 0.666·21-s + (−1 + i)23-s + (−0.280 − 0.959i)25-s + (0.592 + 0.592i)27-s − 0.275i·29-s − 0.451·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.995 + 0.0898i$
motivic weight  =  \(2\)
character  :  $\chi_{20} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :1),\ 0.995 + 0.0898i)$
$L(\frac{3}{2})$  $\approx$  $0.855497 - 0.0384920i$
$L(\frac12)$  $\approx$  $0.855497 - 0.0384920i$
$L(2)$   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 \( 1 \)
5 \( 1 + (3 - 4i)T \)
good3 \( 1 + (-1 + i)T - 9iT^{2} \)
7 \( 1 + (7 + 7i)T + 49iT^{2} \)
11 \( 1 - 10T + 121T^{2} \)
13 \( 1 + (-9 + 9i)T - 169iT^{2} \)
17 \( 1 + (-1 - i)T + 289iT^{2} \)
19 \( 1 + 8iT - 361T^{2} \)
23 \( 1 + (23 - 23i)T - 529iT^{2} \)
29 \( 1 + 8iT - 841T^{2} \)
31 \( 1 + 14T + 961T^{2} \)
37 \( 1 + (-33 - 33i)T + 1.36e3iT^{2} \)
41 \( 1 + 14T + 1.68e3T^{2} \)
43 \( 1 + (15 - 15i)T - 1.84e3iT^{2} \)
47 \( 1 + (39 + 39i)T + 2.20e3iT^{2} \)
53 \( 1 + (7 - 7i)T - 2.80e3iT^{2} \)
59 \( 1 + 56iT - 3.48e3T^{2} \)
61 \( 1 - 42T + 3.72e3T^{2} \)
67 \( 1 + (7 + 7i)T + 4.48e3iT^{2} \)
71 \( 1 - 98T + 5.04e3T^{2} \)
73 \( 1 + (-49 + 49i)T - 5.32e3iT^{2} \)
79 \( 1 - 96iT - 6.24e3T^{2} \)
83 \( 1 + (63 - 63i)T - 6.88e3iT^{2} \)
89 \( 1 + 112iT - 7.92e3T^{2} \)
97 \( 1 + (-33 - 33i)T + 9.40e3iT^{2} \)
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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

−18.34748371195793641274209620802, −16.79087467384288582269870954105, −15.61439987312263760175240269667, −14.10552965086306807216091377726, −13.13258781276970670552049700998, −11.32836035526420098193819171841, −10.02152727589583107599767457175, −7.931175664981236379641946395835, −6.64188074165720645380338848513, −3.58149732306835412927525427068, 3.85769210437284222679930271984, 6.27462099292704690001749659643, 8.664380106294927574380010983587, 9.494964812819170688883265412784, 11.78936999005442889405940598420, 12.68128665727435992573424581356, 14.50339807607041166725982042675, 15.77141160655571916844795700460, 16.54259600717954489917133610160, 18.35124538472176511757358714269

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