Properties

Degree $2$
Conductor $5577$
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 + 2·5-s + 6-s − 3·8-s + 9-s + 2·10-s + 11-s − 12-s + 2·15-s − 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s + 22-s − 8·23-s − 3·24-s − 25-s + 27-s − 10·29-s + 2·30-s + 5·32-s + 33-s − 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.365·30-s + 0.883·32-s + 0.174·33-s − 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5577\)    =    \(3 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{5577} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−17.73205709195094, −17.55920996585739, −16.65821421740992, −15.74436277765846, −15.47530333104362, −14.44762864442917, −14.27026947279192, −13.67677016178829, −13.06422246977793, −12.83531062066030, −11.74902270830596, −11.38382810323156, −10.22358122690508, −9.660401895548503, −9.303658983572537, −8.522626716892774, −7.938587060470884, −6.848465552902963, −6.299622277677048, −5.455252498966306, −4.960185992984079, −3.893668447508626, −3.557790488853923, −2.346749198016473, −1.723171728658404, 0, 1.723171728658404, 2.346749198016473, 3.557790488853923, 3.893668447508626, 4.960185992984079, 5.455252498966306, 6.299622277677048, 6.848465552902963, 7.938587060470884, 8.522626716892774, 9.303658983572537, 9.660401895548503, 10.22358122690508, 11.38382810323156, 11.74902270830596, 12.83531062066030, 13.06422246977793, 13.67677016178829, 14.27026947279192, 14.44762864442917, 15.47530333104362, 15.74436277765846, 16.65821421740992, 17.55920996585739, 17.73205709195094

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