Properties

Degree $2$
Conductor $5610$
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 − 8-s + 9-s + 10-s + 11-s + 12-s − 2·13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 8·23-s − 24-s + 25-s + 2·26-s + 27-s − 6·29-s + 30-s − 8·31-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 − 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.223·20-s − 0.213·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.57760691531422, −17.50188565474811, −16.60121009502069, −16.15088277366414, −15.41231567580489, −14.90485849010456, −14.47647455567912, −13.65994101166357, −12.87366079147875, −12.42248362025048, −11.65693295783545, −10.89784215087936, −10.59268143776993, −9.458678276889811, −9.218797902218057, −8.580295749115586, −7.778593550053777, −7.187797553433692, −6.779285690725873, −5.626842201287031, −4.880406240670986, −3.858363209105181, −3.199921271600588, −2.262256150802769, −1.340661552420221, 0, 1.340661552420221, 2.262256150802769, 3.199921271600588, 3.858363209105181, 4.880406240670986, 5.626842201287031, 6.779285690725873, 7.187797553433692, 7.778593550053777, 8.580295749115586, 9.218797902218057, 9.458678276889811, 10.59268143776993, 10.89784215087936, 11.65693295783545, 12.42248362025048, 12.87366079147875, 13.65994101166357, 14.47647455567912, 14.90485849010456, 15.41231567580489, 16.15088277366414, 16.60121009502069, 17.50188565474811, 17.57760691531422

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