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 + 4·5-s + 6-s − 2·7-s + 8-s + 9-s + 4·10-s + 11-s + 12-s − 2·14-s + 4·15-s + 16-s − 17-s + 18-s + 4·20-s − 2·21-s + 22-s − 6·23-s + 24-s + 11·25-s + 27-s − 2·28-s − 2·29-s + 4·30-s + 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.894·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.730·30-s + 0.718·31-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

\(L(1)\) \(\approx\) \(3.827933235\)
\(L(\frac12)\) \(\approx\) \(3.827933235\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−19.48484011569538, −18.46192536321198, −17.95716393510138, −17.05010588842074, −16.50555223403942, −15.68208021244174, −14.81685355416984, −14.19732053129507, −13.54866702540893, −13.20150895181096, −12.44455790920466, −11.50293441178465, −10.25516847678640, −9.957781387417194, −9.184187584794878, −8.288845053216673, −7.000373650045399, −6.327651793952685, −5.711403571184931, −4.664355373423064, −3.480479945649647, −2.519433271965645, −1.653973356462181, 1.653973356462181, 2.519433271965645, 3.480479945649647, 4.664355373423064, 5.711403571184931, 6.327651793952685, 7.000373650045399, 8.288845053216673, 9.184187584794878, 9.957781387417194, 10.25516847678640, 11.50293441178465, 12.44455790920466, 13.20150895181096, 13.54866702540893, 14.19732053129507, 14.81685355416984, 15.68208021244174, 16.50555223403942, 17.05010588842074, 17.95716393510138, 18.46192536321198, 19.48484011569538

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