Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \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 + 5-s − 2·7-s + 9-s + 2·13-s + 2·15-s − 4·19-s − 4·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s − 10·43-s + 45-s − 6·47-s − 3·49-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s − 2·63-s + 2·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5780\)    =    \(2^{2} \cdot 5 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5780} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5780,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - T \)
17 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−17.89376717008518, −17.08131477745705, −16.65231917376241, −15.84423209266942, −15.40549027220513, −14.65209191137663, −14.27361157902326, −13.43872061026826, −13.27049977695806, −12.57144750482827, −11.71518947135603, −11.06118665359043, −10.04318080877735, −9.859256015272710, −9.067859604166774, −8.379366974510093, −8.082512830031549, −7.016855783184379, −6.382035091812447, −5.765072776007429, −4.759217719418894, −3.705474005098914, −3.344516011166326, −2.326936002339414, −1.698704203094087, 0, 1.698704203094087, 2.326936002339414, 3.344516011166326, 3.705474005098914, 4.759217719418894, 5.765072776007429, 6.382035091812447, 7.016855783184379, 8.082512830031549, 8.379366974510093, 9.067859604166774, 9.859256015272710, 10.04318080877735, 11.06118665359043, 11.71518947135603, 12.57144750482827, 13.27049977695806, 13.43872061026826, 14.27361157902326, 14.65209191137663, 15.40549027220513, 15.84423209266942, 16.65231917376241, 17.08131477745705, 17.89376717008518

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