Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.896 - 0.443i$
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 + (−0.278 − 0.142i)13-s + (−1.76 − 0.278i)17-s + (−1.11 + 1.53i)29-s + (−0.412 + 0.809i)37-s + (0.363 + 1.11i)41-s i·49-s + (−1.95 + 0.309i)53-s + (−0.363 + 1.11i)61-s + (−0.896 − 1.76i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)9-s + (−0.278 − 0.142i)13-s + (−1.76 − 0.278i)17-s + (−1.11 + 1.53i)29-s + (−0.412 + 0.809i)37-s + (0.363 + 1.11i)41-s i·49-s + (−1.95 + 0.309i)53-s + (−0.363 + 1.11i)61-s + (−0.896 − 1.76i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.896 - 0.443i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ -0.896 - 0.443i)$
$L(\frac{1}{2})$  $\approx$  $0.2853206794$
$L(\frac12)$  $\approx$  $0.2853206794$
$L(1)$   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 \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
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 \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
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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

−8.962016502698125478465909867741, −8.271039984676221234090137167219, −7.49702533751218186170909572541, −6.71169160202108214149234666725, −6.04152948710413406921592244264, −5.11091042815852925073194855450, −4.58873938611147213913193441051, −3.42625349440374281574296387389, −2.66769380532270550784370097460, −1.69603992315337827643740621500, 0.14373568290828362381687220266, 1.93138120635499229201319970850, 2.66886965184114505833188087916, 3.75812110907431574083513251902, 4.45094088031487604052914311015, 5.41848009467673031714237459598, 6.13225850348543145384816549660, 6.75136905827156665914065291522, 7.66303704782819474545662029096, 8.322610863824198569568538997803

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