Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26·3-s − 22·7-s + 433·9-s + 768·11-s + 46·13-s − 378·17-s − 1.10e3·19-s + 572·21-s − 1.98e3·23-s − 4.94e3·27-s − 5.61e3·29-s + 3.98e3·31-s − 1.99e4·33-s + 142·37-s − 1.19e3·39-s + 1.54e3·41-s − 5.02e3·43-s + 2.47e4·47-s − 1.63e4·49-s + 9.82e3·51-s + 1.41e4·53-s + 2.86e4·57-s − 2.83e4·59-s + 5.52e3·61-s − 9.52e3·63-s − 2.47e4·67-s + 5.16e4·69-s + ⋯
L(s)  = 1  − 1.66·3-s − 0.169·7-s + 1.78·9-s + 1.91·11-s + 0.0754·13-s − 0.317·17-s − 0.699·19-s + 0.283·21-s − 0.782·23-s − 1.30·27-s − 1.23·29-s + 0.745·31-s − 3.19·33-s + 0.0170·37-s − 0.125·39-s + 0.143·41-s − 0.414·43-s + 1.63·47-s − 0.971·49-s + 0.529·51-s + 0.692·53-s + 1.16·57-s − 1.06·59-s + 0.190·61-s − 0.302·63-s − 0.673·67-s + 1.30·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{400} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 400,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $1.012626024$
$L(\frac12)$  $\approx$  $1.012626024$
$L(\frac{7}{2})$   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 \( 1 + 26 T + p^{5} T^{2} \)
7 \( 1 + 22 T + p^{5} T^{2} \)
11 \( 1 - 768 T + p^{5} T^{2} \)
13 \( 1 - 46 T + p^{5} T^{2} \)
17 \( 1 + 378 T + p^{5} T^{2} \)
19 \( 1 + 1100 T + p^{5} T^{2} \)
23 \( 1 + 1986 T + p^{5} T^{2} \)
29 \( 1 + 5610 T + p^{5} T^{2} \)
31 \( 1 - 3988 T + p^{5} T^{2} \)
37 \( 1 - 142 T + p^{5} T^{2} \)
41 \( 1 - 1542 T + p^{5} T^{2} \)
43 \( 1 + 5026 T + p^{5} T^{2} \)
47 \( 1 - 24738 T + p^{5} T^{2} \)
53 \( 1 - 14166 T + p^{5} T^{2} \)
59 \( 1 + 28380 T + p^{5} T^{2} \)
61 \( 1 - 5522 T + p^{5} T^{2} \)
67 \( 1 + 24742 T + p^{5} T^{2} \)
71 \( 1 + 42372 T + p^{5} T^{2} \)
73 \( 1 - 52126 T + p^{5} T^{2} \)
79 \( 1 - 39640 T + p^{5} T^{2} \)
83 \( 1 + 59826 T + p^{5} T^{2} \)
89 \( 1 - 57690 T + p^{5} T^{2} \)
97 \( 1 - 144382 T + p^{5} T^{2} \)
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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

−10.66049697553484844364589976630, −9.722829854459598029461233029726, −8.797512192057952053313099815564, −7.30103624781289634156379294086, −6.38477507214835223463864881094, −5.94135384705747554999519759253, −4.63348310676988706331941361674, −3.82671477962914164662221660740, −1.73680774699113484770365600003, −0.59847245558727495978595998991, 0.59847245558727495978595998991, 1.73680774699113484770365600003, 3.82671477962914164662221660740, 4.63348310676988706331941361674, 5.94135384705747554999519759253, 6.38477507214835223463864881094, 7.30103624781289634156379294086, 8.797512192057952053313099815564, 9.722829854459598029461233029726, 10.66049697553484844364589976630

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