Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·3-s + 158·7-s − 47·9-s + 148·11-s − 684·13-s − 2.04e3·17-s − 2.22e3·19-s + 2.21e3·21-s − 1.24e3·23-s − 4.06e3·27-s − 270·29-s + 2.04e3·31-s + 2.07e3·33-s + 4.37e3·37-s − 9.57e3·39-s − 2.39e3·41-s + 2.29e3·43-s − 1.06e4·47-s + 8.15e3·49-s − 2.86e4·51-s − 2.96e3·53-s − 3.10e4·57-s + 3.97e4·59-s − 4.22e4·61-s − 7.42e3·63-s + 3.20e4·67-s − 1.74e4·69-s + ⋯
L(s)  = 1  + 0.898·3-s + 1.21·7-s − 0.193·9-s + 0.368·11-s − 1.12·13-s − 1.71·17-s − 1.41·19-s + 1.09·21-s − 0.491·23-s − 1.07·27-s − 0.0596·29-s + 0.382·31-s + 0.331·33-s + 0.525·37-s − 1.00·39-s − 0.222·41-s + 0.189·43-s − 0.705·47-s + 0.485·49-s − 1.54·51-s − 0.144·53-s − 1.26·57-s + 1.48·59-s − 1.45·61-s − 0.235·63-s + 0.873·67-s − 0.441·69-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  =  \(1\)
Selberg data  =  \((2,\ 400,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   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 \( 1 - 14 T + p^{5} T^{2} \)
7 \( 1 - 158 T + p^{5} T^{2} \)
11 \( 1 - 148 T + p^{5} T^{2} \)
13 \( 1 + 684 T + p^{5} T^{2} \)
17 \( 1 + 2048 T + p^{5} T^{2} \)
19 \( 1 + 2220 T + p^{5} T^{2} \)
23 \( 1 + 1246 T + p^{5} T^{2} \)
29 \( 1 + 270 T + p^{5} T^{2} \)
31 \( 1 - 2048 T + p^{5} T^{2} \)
37 \( 1 - 4372 T + p^{5} T^{2} \)
41 \( 1 + 2398 T + p^{5} T^{2} \)
43 \( 1 - 2294 T + p^{5} T^{2} \)
47 \( 1 + 10682 T + p^{5} T^{2} \)
53 \( 1 + 2964 T + p^{5} T^{2} \)
59 \( 1 - 39740 T + p^{5} T^{2} \)
61 \( 1 + 42298 T + p^{5} T^{2} \)
67 \( 1 - 32098 T + p^{5} T^{2} \)
71 \( 1 - 4248 T + p^{5} T^{2} \)
73 \( 1 + 30104 T + p^{5} T^{2} \)
79 \( 1 + 35280 T + p^{5} T^{2} \)
83 \( 1 + 27826 T + p^{5} T^{2} \)
89 \( 1 + 85210 T + p^{5} T^{2} \)
97 \( 1 - 97232 T + p^{5} T^{2} \)
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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

−9.884620709906532758768957842059, −8.817228191598518514928911629022, −8.352345148345548116621176433427, −7.39943138420528753341728918011, −6.27855085331064434988939796234, −4.86364865258870609673186413208, −4.10701862125027924524668219856, −2.54371976867165853289592280491, −1.86816374927412047878431731771, 0, 1.86816374927412047878431731771, 2.54371976867165853289592280491, 4.10701862125027924524668219856, 4.86364865258870609673186413208, 6.27855085331064434988939796234, 7.39943138420528753341728918011, 8.352345148345548116621176433427, 8.817228191598518514928911629022, 9.884620709906532758768957842059

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