Properties

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

Origins

Related objects

Downloads

Learn more about

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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.797 - 0.603i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.797 - 0.603i)$
$L(\frac{1}{2})$  $\approx$  $1.219460150$
$L(\frac12)$  $\approx$  $1.219460150$
$L(1)$   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 \( 1 + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.685562029224743100475070326393, −7.964543406605789324298236769241, −7.36794006423324002193215077861, −6.45531679637615261693075260807, −5.50354317954882683202894764082, −5.31388922453337902750425552965, −3.97119494904179707700201623494, −3.31183418884635302665518492425, −2.37278246452391450312378259737, −1.17462569582701276217779496759, 0.796156898929570856522469151921, 2.18089425945419952803428963485, 3.16506333841467406646810723115, 3.79140280183946299740506364236, 4.88313122617080892124306144471, 5.81571222009704030093654407645, 6.02969586048932757912666489954, 7.24261362683337572403111034875, 7.84057033884969017682353192431, 8.478885472324683250174087789972

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