Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s − 13-s − 4·19-s − 21-s + 5·27-s + 6·29-s + 31-s − 2·37-s + 39-s + 2·43-s − 6·47-s − 6·49-s + 3·53-s + 4·57-s − 6·59-s + 10·61-s − 2·63-s − 4·67-s + 3·71-s − 2·73-s + 79-s + 81-s + 12·83-s − 6·87-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.277·13-s − 0.917·19-s − 0.218·21-s + 0.962·27-s + 1.11·29-s + 0.179·31-s − 0.328·37-s + 0.160·39-s + 0.304·43-s − 0.875·47-s − 6/7·49-s + 0.412·53-s + 0.529·57-s − 0.781·59-s + 1.28·61-s − 0.251·63-s − 0.488·67-s + 0.356·71-s − 0.234·73-s + 0.112·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{28900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 28900,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.41190993680490, −14.92305753516091, −14.29007103841956, −14.06416324211187, −13.25066739102567, −12.73700403205724, −12.14799436045205, −11.72002272875549, −11.17856917138163, −10.69609733378137, −10.16806838570428, −9.569117740876812, −8.734976020031851, −8.449530375341181, −7.817152534779036, −7.097839295340376, −6.374915277588256, −6.116612346759028, −5.198330540547061, −4.872913193988328, −4.170978468263983, −3.318361964187050, −2.618353860016297, −1.890163855714809, −0.9024273072825576, 0, 0.9024273072825576, 1.890163855714809, 2.618353860016297, 3.318361964187050, 4.170978468263983, 4.872913193988328, 5.198330540547061, 6.116612346759028, 6.374915277588256, 7.097839295340376, 7.817152534779036, 8.449530375341181, 8.734976020031851, 9.569117740876812, 10.16806838570428, 10.69609733378137, 11.17856917138163, 11.72002272875549, 12.14799436045205, 12.73700403205724, 13.25066739102567, 14.06416324211187, 14.29007103841956, 14.92305753516091, 15.41190993680490

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