Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.603 + 0.797i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.603 + 0.797i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (993, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.603 + 0.797i)$
$L(\frac{1}{2})$  $\approx$  $1.393337988$
$L(\frac12)$  $\approx$  $1.393337988$
$L(1)$   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 \( 1 \)
good3 \( 1 + (-0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.599773307452145536252978837235, −7.61356939748501633282296986458, −7.22967374809209278433798129336, −6.32234070202906041104224057211, −5.44445313270051741933646964242, −4.93413493819491911179388056416, −3.64287339607775174670315998273, −3.32809014689970200764436480556, −1.96819925723893023740312552428, −0.842515910366969655842917993558, 1.48825040359220765606321554022, 2.07116336530858488929950858830, 3.55697033631481768419599102173, 4.12418432105970475336511024371, 4.84451652299861274245209553398, 5.92281261241630302735032277258, 6.56457647911929271893794644833, 7.19510400631460177604893541842, 8.047808839199457332303873745098, 8.714026072837800928149217760289

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