Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
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-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s − 11-s + 12-s − 4·13-s − 2·14-s + 15-s + 16-s + 17-s + 18-s − 6·19-s + 20-s − 2·21-s − 22-s − 2·23-s + 24-s + 25-s − 4·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s − 0.213·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5610} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5610,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−17.71022133542766, −17.19909494171114, −16.43464199803659, −16.11885217867544, −15.15109797380037, −14.81388952629550, −14.33488035710996, −13.52627315830931, −13.01366065577202, −12.64353393517663, −11.99568563640709, −11.08738762096829, −10.41127597747242, −9.791369620204371, −9.309055703012188, −8.401696175915671, −7.707039502152055, −6.931532223376482, −6.406202027119515, −5.547469514232438, −4.884697974924909, −3.980543395872272, −3.287523162984481, −2.419222035352901, −1.817449207142474, 0, 1.817449207142474, 2.419222035352901, 3.287523162984481, 3.980543395872272, 4.884697974924909, 5.547469514232438, 6.406202027119515, 6.931532223376482, 7.707039502152055, 8.401696175915671, 9.309055703012188, 9.791369620204371, 10.41127597747242, 11.08738762096829, 11.99568563640709, 12.64353393517663, 13.01366065577202, 13.52627315830931, 14.33488035710996, 14.81388952629550, 15.15109797380037, 16.11885217867544, 16.43464199803659, 17.19909494171114, 17.71022133542766

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