Properties

Degree $2$
Conductor $17661$
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 + 7-s − 3·8-s + 9-s − 2·10-s − 4·11-s − 12-s − 2·13-s + 14-s − 2·15-s − 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s − 4·22-s − 3·24-s − 25-s − 2·26-s + 27-s − 28-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17661\)    =    \(3 \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{17661} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17661,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
29 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−16.01687889258196, −15.35668423820548, −14.84009161910233, −14.66553413613092, −13.75309976205922, −13.33200217248414, −13.11036404297744, −12.16956827752395, −11.85994937870270, −11.34144723971792, −10.41722200230207, −9.940688679252846, −9.313308864940298, −8.576915823588736, −8.115650110430112, −7.643074253300940, −7.048380748442450, −6.093898649191051, −5.376666146691944, −4.818119595018552, −4.251173887569727, −3.722313208735415, −2.761236310388546, −2.517060379338600, −1.011536520717521, 0, 1.011536520717521, 2.517060379338600, 2.761236310388546, 3.722313208735415, 4.251173887569727, 4.818119595018552, 5.376666146691944, 6.093898649191051, 7.048380748442450, 7.643074253300940, 8.115650110430112, 8.576915823588736, 9.313308864940298, 9.940688679252846, 10.41722200230207, 11.34144723971792, 11.85994937870270, 12.16956827752395, 13.11036404297744, 13.33200217248414, 13.75309976205922, 14.66553413613092, 14.84009161910233, 15.35668423820548, 16.01687889258196

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