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 + 4·19-s + 20-s − 22-s − 4·23-s + 24-s + 25-s − 2·26-s − 27-s − 6·29-s + 30-s − 8·31-s − 32-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 + 0.917·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-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 + ⋯

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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−18.05625634879067, −17.22108382847283, −16.71269936793620, −16.24493293108048, −15.67330816382913, −14.92128979401218, −14.32770335018626, −13.50105276766001, −13.02732810055236, −12.19147790478160, −11.63049202180359, −11.03096588928311, −10.51269356656816, −9.668812760978507, −9.338187976355957, −8.553844039144271, −7.717920399105085, −7.153525770006678, −6.316637745737227, −5.808768063037684, −5.087823347076959, −4.026847431290952, −3.213399065517432, −2.006305559561748, −1.298492104771207, 0, 1.298492104771207, 2.006305559561748, 3.213399065517432, 4.026847431290952, 5.087823347076959, 5.808768063037684, 6.316637745737227, 7.153525770006678, 7.717920399105085, 8.553844039144271, 9.338187976355957, 9.668812760978507, 10.51269356656816, 11.03096588928311, 11.63049202180359, 12.19147790478160, 13.02732810055236, 13.50105276766001, 14.32770335018626, 14.92128979401218, 15.67330816382913, 16.24493293108048, 16.71269936793620, 17.22108382847283, 18.05625634879067

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