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  + 2·3-s + 2·7-s + 9-s − 5·11-s − 6·13-s + 7·19-s + 4·21-s + 2·23-s − 4·27-s + 9·29-s − 8·31-s − 10·33-s − 12·39-s − 7·41-s + 10·43-s + 8·47-s − 3·49-s + 4·53-s + 14·57-s + 9·59-s + 5·61-s + 2·63-s + 4·67-s + 4·69-s − 71-s − 4·73-s − 10·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 1.66·13-s + 1.60·19-s + 0.872·21-s + 0.417·23-s − 0.769·27-s + 1.67·29-s − 1.43·31-s − 1.74·33-s − 1.92·39-s − 1.09·41-s + 1.52·43-s + 1.16·47-s − 3/7·49-s + 0.549·53-s + 1.85·57-s + 1.17·59-s + 0.640·61-s + 0.251·63-s + 0.488·67-s + 0.481·69-s − 0.118·71-s − 0.468·73-s − 1.13·77-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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.32636049173909, −14.85520671266958, −14.23841693208610, −14.12008152947377, −13.43142596173638, −12.88348148422447, −12.31429000114887, −11.77247497596427, −11.14182645744634, −10.52358791409601, −9.843632953297810, −9.624382664039723, −8.747768860147224, −8.370277839628524, −7.754400742789737, −7.326381123895950, −7.002967284548155, −5.548538603278196, −5.402736636235321, −4.744639676894113, −4.001680377769927, −3.053065592357211, −2.669633884753383, −2.194840708588074, −1.171984512583126, 0, 1.171984512583126, 2.194840708588074, 2.669633884753383, 3.053065592357211, 4.001680377769927, 4.744639676894113, 5.402736636235321, 5.548538603278196, 7.002967284548155, 7.326381123895950, 7.754400742789737, 8.370277839628524, 8.747768860147224, 9.624382664039723, 9.843632953297810, 10.52358791409601, 11.14182645744634, 11.77247497596427, 12.31429000114887, 12.88348148422447, 13.43142596173638, 14.12008152947377, 14.23841693208610, 14.85520671266958, 15.32636049173909

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