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 + 6·29-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 + 1.11·29-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

\(L(1)\) \(\approx\) \(3.366050137\)
\(L(\frac12)\) \(\approx\) \(3.366050137\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−19.41492992746137, −18.70218208638716, −17.76906366590727, −17.24284220260120, −16.31142522978955, −15.56933896107460, −14.91790943896770, −14.30223710062155, −13.77986423090923, −13.02659278596736, −12.05720738747138, −11.64830738299953, −10.70967834567625, −9.752369856377215, −9.094379826439675, −7.935038583530896, −7.431324671549312, −6.527479894672070, −5.224691283914207, −4.767523763903929, −3.565584005096202, −2.663333007344193, −1.459631116376295, 1.459631116376295, 2.663333007344193, 3.565584005096202, 4.767523763903929, 5.224691283914207, 6.527479894672070, 7.431324671549312, 7.935038583530896, 9.094379826439675, 9.752369856377215, 10.70967834567625, 11.64830738299953, 12.05720738747138, 13.02659278596736, 13.77986423090923, 14.30223710062155, 14.91790943896770, 15.56933896107460, 16.31142522978955, 17.24284220260120, 17.76906366590727, 18.70218208638716, 19.41492992746137

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