Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.936 + 0.349i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)9-s + (−0.142 + 0.896i)13-s + (0.278 − 0.142i)17-s + (1.11 − 0.363i)29-s + (1.95 + 0.309i)37-s + (1.53 − 1.11i)41-s i·49-s + (1.58 + 0.809i)53-s + (−1.53 − 1.11i)61-s + (−1.76 + 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (0.896 − 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)9-s + (−0.142 + 0.896i)13-s + (0.278 − 0.142i)17-s + (1.11 − 0.363i)29-s + (1.95 + 0.309i)37-s + (1.53 − 1.11i)41-s i·49-s + (1.58 + 0.809i)53-s + (−1.53 − 1.11i)61-s + (−1.76 + 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (0.896 − 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.936 + 0.349i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (3457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.936 + 0.349i)$
$L(\frac{1}{2})$  $\approx$  $1.243683497$
$L(\frac12)$  $\approx$  $1.243683497$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.95 - 0.309i)T + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-1.58 - 0.809i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
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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

−8.768936956863952426856705783828, −7.81290366635244655285951402117, −7.14166683074017238291324011103, −6.25159723909029149179616191432, −5.84094598483132373083210550006, −4.70889562720184078455647837515, −4.05922371059531065298397089313, −3.08297345838531656918322175391, −2.25216586337053787055338854721, −0.876628085897858628262699573574, 1.06279979876293503187293731204, 2.48985672056177143594054329864, 2.99578827622824067597478935302, 4.21005358235834226903620952262, 4.92717321185023965981026197802, 5.77435490952757076617706843744, 6.27392303197630440052101944047, 7.50408943150717824527051177474, 7.82050157150394267143334467161, 8.600612902783382737166569245059

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