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 + 4·7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 4·13-s − 4·14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s − 20-s − 4·21-s − 22-s + 24-s + 25-s + 4·26-s − 27-s + 4·28-s − 8·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-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 − 1.06·14-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 − 0.872·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.755·28-s − 1.48·29-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−17.71039909782316, −17.26858363363054, −16.89011654108891, −16.22200295105286, −15.38528598611805, −15.06852071348127, −14.30086994073014, −13.84604467231057, −12.70828079508954, −12.05081641424193, −11.75108222050162, −11.02696934617482, −10.68162244455878, −9.705703895892917, −9.273517338430474, −8.296698472046037, −7.774051400084104, −7.303413344149271, −6.563230807877515, −5.432699086174819, −5.065921342822094, −4.213315549220850, −3.190005858294994, −1.984964959014038, −1.279273937912346, 0, 1.279273937912346, 1.984964959014038, 3.190005858294994, 4.213315549220850, 5.065921342822094, 5.432699086174819, 6.563230807877515, 7.303413344149271, 7.774051400084104, 8.296698472046037, 9.273517338430474, 9.705703895892917, 10.68162244455878, 11.02696934617482, 11.75108222050162, 12.05081641424193, 12.70828079508954, 13.84604467231057, 14.30086994073014, 15.06852071348127, 15.38528598611805, 16.22200295105286, 16.89011654108891, 17.26858363363054, 17.71039909782316

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