Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.612 - 0.790i$
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 + (−0.278 + 1.76i)13-s + (−1.76 + 0.896i)17-s + (−1.11 + 0.363i)29-s + (−0.309 − 0.0489i)37-s + (−1.53 + 1.11i)41-s i·49-s + (0.809 + 0.412i)53-s + (1.53 + 1.11i)61-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (−0.142 + 0.278i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)9-s + (−0.278 + 1.76i)13-s + (−1.76 + 0.896i)17-s + (−1.11 + 0.363i)29-s + (−0.309 − 0.0489i)37-s + (−1.53 + 1.11i)41-s i·49-s + (0.809 + 0.412i)53-s + (1.53 + 1.11i)61-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (−0.142 + 0.278i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.612 - 0.790i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (3457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ -0.612 - 0.790i)$
$L(\frac{1}{2})$  $\approx$  $0.5454210538$
$L(\frac12)$  $\approx$  $0.5454210538$
$L(1)$   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 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
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 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
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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

−8.780312714046468729201109606237, −8.486165683131870647605389181444, −7.18106959787945224830752980225, −6.70600394552555355152902566726, −6.11201174933033780732850014524, −5.12433608789207800505249670657, −4.21569492320499818761247706300, −3.67468320598989029457680935535, −2.44437199663304278581876360370, −1.63611253974029431205156335384, 0.27872918691562738619872904327, 2.04503085101582213803535825315, 2.74425083124120998568537263717, 3.66515482966975001347982213215, 4.80685494652666614713014052584, 5.29817380628038188944813623945, 6.04787870666408704959346822591, 7.06405306828222902180341786055, 7.60365246983964020145500405557, 8.445785664905282110209859723514

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