Properties

Degree $2$
Conductor $5610$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 11-s + 12-s + 4·13-s − 15-s + 16-s + 17-s − 18-s − 8·19-s − 20-s + 22-s + 4·23-s − 24-s + 25-s − 4·26-s + 27-s + 30-s − 6·31-s − 32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.182·30-s − 1.07·31-s − 0.176·32-s − 0.174·33-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

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{5610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.99670820994760, −17.15923449246211, −16.73808753170263, −15.97717163887649, −15.55421861685639, −14.91681746176655, −14.45496197154102, −13.57333417453953, −12.91918240898336, −12.50635779750253, −11.56880878313838, −10.89457759478496, −10.56375482650968, −9.750065106278679, −8.815831873598253, −8.648130370969623, −7.964530061005053, −7.189198285055140, −6.597683252937249, −5.790111875607132, −4.790897836360178, −3.856087754551494, −3.244572240827828, −2.224779217959230, −1.357623153637054, 0, 1.357623153637054, 2.224779217959230, 3.244572240827828, 3.856087754551494, 4.790897836360178, 5.790111875607132, 6.597683252937249, 7.189198285055140, 7.964530061005053, 8.648130370969623, 8.815831873598253, 9.750065106278679, 10.56375482650968, 10.89457759478496, 11.56880878313838, 12.50635779750253, 12.91918240898336, 13.57333417453953, 14.45496197154102, 14.91681746176655, 15.55421861685639, 15.97717163887649, 16.73808753170263, 17.15923449246211, 17.99670820994760

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