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 + 6-s + 2·7-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 2·14-s + 16-s − 17-s + 18-s − 8·19-s + 2·21-s + 22-s + 6·23-s + 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s − 2·29-s + 4·31-s + 32-s + 33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.174·33-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

\(L(1)\) \(\approx\) \(3.429147609\)
\(L(\frac12)\) \(\approx\) \(3.429147609\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−19.20789055728551, −18.89088382121183, −17.70320396919337, −17.28740714649915, −16.27871160973339, −15.58963999100044, −14.83539955812831, −14.51632244000988, −13.47185593059453, −13.18330534052198, −12.25473677331962, −11.29568449574900, −10.88060767906729, −9.891765499192152, −8.672521011063549, −8.394919814891872, −7.243077874064986, −6.450791931926676, −5.517242207641230, −4.395653432484006, −3.814054297507792, −2.549662821755409, −1.498836925426366, 1.498836925426366, 2.549662821755409, 3.814054297507792, 4.395653432484006, 5.517242207641230, 6.450791931926676, 7.243077874064986, 8.394919814891872, 8.672521011063549, 9.891765499192152, 10.88060767906729, 11.29568449574900, 12.25473677331962, 13.18330534052198, 13.47185593059453, 14.51632244000988, 14.83539955812831, 15.58963999100044, 16.27871160973339, 17.28740714649915, 17.70320396919337, 18.89088382121183, 19.20789055728551

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