Properties

Label 2-1122-1.1-c1-0-12
Degree $2$
Conductor $1122$
Sign $1$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·7-s + 8-s + 9-s + 2·10-s − 11-s + 12-s + 4·13-s − 2·14-s + 2·15-s + 16-s + 17-s + 18-s + 2·19-s + 2·20-s − 2·21-s − 22-s + 2·23-s + 24-s − 25-s + 4·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.447·20-s − 0.436·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-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: $1$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1122} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.456030932\)
\(L(\frac12)\) \(\approx\) \(3.456030932\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.721493808888347744652427116171, −9.208514501966211369077275737330, −8.137355495831918287122633547549, −7.23609290526876479191984821255, −6.21041754054960026282024258092, −5.76385775437179457867723695994, −4.56177858452333445614637555384, −3.46523248766531211326925164398, −2.72892123253348569303169805798, −1.47923983647316370261704285245, 1.47923983647316370261704285245, 2.72892123253348569303169805798, 3.46523248766531211326925164398, 4.56177858452333445614637555384, 5.76385775437179457867723695994, 6.21041754054960026282024258092, 7.23609290526876479191984821255, 8.137355495831918287122633547549, 9.208514501966211369077275737330, 9.721493808888347744652427116171

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