Properties

Degree $2$
Conductor $1122$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 2·5-s + 6-s + 4·7-s − 8-s + 9-s − 2·10-s + 11-s − 12-s + 6·13-s − 4·14-s − 2·15-s + 16-s − 17-s − 18-s + 4·19-s + 2·20-s − 4·21-s − 22-s + 24-s − 25-s − 6·26-s − 27-s + 4·28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.872·21-s − 0.213·22-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.755·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1122} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.512218793\)
\(L(\frac12)\) \(\approx\) \(1.512218793\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−18.93368673565616, −18.23314101088264, −17.96310476107531, −17.29513571678291, −16.81420605511911, −15.77893426260094, −15.36734820059939, −14.08647278653550, −13.88763515234892, −12.85030938645632, −11.56591956481005, −11.53794384257579, −10.53304772003330, −9.990310296989423, −8.841178745369806, −8.428954650442548, −7.380631418578447, −6.461754087582877, −5.658559883357037, −4.908843061052333, −3.580983834355331, −1.890185381135888, −1.238961368648041, 1.238961368648041, 1.890185381135888, 3.580983834355331, 4.908843061052333, 5.658559883357037, 6.461754087582877, 7.380631418578447, 8.428954650442548, 8.841178745369806, 9.990310296989423, 10.53304772003330, 11.53794384257579, 11.56591956481005, 12.85030938645632, 13.88763515234892, 14.08647278653550, 15.36734820059939, 15.77893426260094, 16.81420605511911, 17.29513571678291, 17.96310476107531, 18.23314101088264, 18.93368673565616

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