Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 28900,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.870743333$
$L(\frac12)$  $\approx$  $1.870743333$
$L(\frac{3}{2})$   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

−15.26530303097837, −14.50187892759277, −14.20630320021564, −13.67094271787825, −13.07123180057918, −12.64066447760423, −12.00317887960638, −11.41053436740471, −10.90620744652468, −10.29150955937782, −9.713444644874046, −9.160042083573306, −8.534339368858225, −8.237091579603191, −7.594951943940518, −6.756236727759150, −6.352345364211605, −5.710237521514620, −5.011375166188910, −4.249171266532877, −3.655247533659185, −2.909375697012957, −2.456992874934301, −1.571189509146740, −0.4998662878130997, 0.4998662878130997, 1.571189509146740, 2.456992874934301, 2.909375697012957, 3.655247533659185, 4.249171266532877, 5.011375166188910, 5.710237521514620, 6.352345364211605, 6.756236727759150, 7.594951943940518, 8.237091579603191, 8.534339368858225, 9.160042083573306, 9.713444644874046, 10.29150955937782, 10.90620744652468, 11.41053436740471, 12.00317887960638, 12.64066447760423, 13.07123180057918, 13.67094271787825, 14.20630320021564, 14.50187892759277, 15.26530303097837

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