Properties

Degree $2$
Conductor $11271$
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 + 3·8-s + 9-s − 2·10-s − 4·11-s − 12-s + 13-s + 2·15-s − 16-s − 18-s − 4·19-s − 2·20-s + 4·22-s + 3·24-s − 25-s − 26-s + 27-s + 2·29-s − 2·30-s + 8·31-s − 5·32-s − 4·33-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.516·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.612·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.43·31-s − 0.883·32-s − 0.696·33-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11271 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11271 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11271\)    =    \(3 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{11271} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11271,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.92759006575354, −16.29252078491741, −15.54010544421274, −15.19857935893781, −14.27848299873055, −13.78649462824098, −13.51060905857509, −12.85660907255773, −12.42644430543789, −11.33746447320516, −10.64356442677509, −10.16766762030403, −9.793274553864360, −9.108835690051499, −8.553604549218620, −7.994979592848307, −7.577486557368084, −6.542493072761149, −6.009492115123270, −5.029036703266981, −4.636301975743210, −3.689856633952203, −2.739647266000850, −2.065617490277759, −1.207684001974968, 0, 1.207684001974968, 2.065617490277759, 2.739647266000850, 3.689856633952203, 4.636301975743210, 5.029036703266981, 6.009492115123270, 6.542493072761149, 7.577486557368084, 7.994979592848307, 8.553604549218620, 9.108835690051499, 9.793274553864360, 10.16766762030403, 10.64356442677509, 11.33746447320516, 12.42644430543789, 12.85660907255773, 13.51060905857509, 13.78649462824098, 14.27848299873055, 15.19857935893781, 15.54010544421274, 16.29252078491741, 16.92759006575354

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