Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 11-s − 12-s − 4·13-s + 2·15-s + 16-s − 17-s + 18-s − 4·19-s − 2·20-s + 22-s − 4·23-s − 24-s − 25-s − 4·26-s − 27-s + 2·30-s + 2·31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-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 + 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.365·30-s + 0.359·31-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: \(1\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.44417026443654, −19.31983122989618, −18.11018778686972, −17.46296394534996, −16.63402118589914, −16.16356590329212, −15.27743463217699, −14.85508453182915, −14.04446599648523, −13.06675060053978, −12.45607493997474, −11.74848178245234, −11.34950568126755, −10.36145381254400, −9.639162665878124, −8.360098088001026, −7.638008145309679, −6.762746427349762, −6.048710512257626, −4.879281563105184, −4.331739502007188, −3.312829525289267, −1.954149571369492, 0, 1.954149571369492, 3.312829525289267, 4.331739502007188, 4.879281563105184, 6.048710512257626, 6.762746427349762, 7.638008145309679, 8.360098088001026, 9.639162665878124, 10.36145381254400, 11.34950568126755, 11.74848178245234, 12.45607493997474, 13.06675060053978, 14.04446599648523, 14.85508453182915, 15.27743463217699, 16.16356590329212, 16.63402118589914, 17.46296394534996, 18.11018778686972, 19.31983122989618, 19.44417026443654

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