Properties

Label 2-11550-1.1-c1-0-24
Degree $2$
Conductor $11550$
Sign $1$
Analytic cond. $92.2272$
Root an. cond. $9.60350$
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 + 6-s + 7-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 14-s + 16-s − 2·17-s + 18-s + 21-s + 22-s − 6·23-s + 24-s + 4·26-s + 27-s + 28-s + 2·31-s + 32-s + 33-s − 2·34-s + 36-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.218·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.174·33-s − 0.342·34-s + 1/6·36-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(92.2272\)
Root analytic conductor: \(9.60350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.157788215\)
\(L(\frac12)\) \(\approx\) \(5.157788215\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.12429654313971, −15.83090132315958, −15.24104587786780, −14.60798842878243, −14.13424662007286, −13.63425896942195, −13.19620844716832, −12.45927544048864, −11.97666131275897, −11.15208868576681, −10.91486347961629, −10.03096090796651, −9.417068112702233, −8.693278019683341, −8.127191991662260, −7.586514905901671, −6.765667105534746, −6.122805819378056, −5.624726466177129, −4.561265673093558, −4.135848723613738, −3.486045515768334, −2.584963225724611, −1.894994249818825, −0.9452887287106955, 0.9452887287106955, 1.894994249818825, 2.584963225724611, 3.486045515768334, 4.135848723613738, 4.561265673093558, 5.624726466177129, 6.122805819378056, 6.765667105534746, 7.586514905901671, 8.127191991662260, 8.693278019683341, 9.417068112702233, 10.03096090796651, 10.91486347961629, 11.15208868576681, 11.97666131275897, 12.45927544048864, 13.19620844716832, 13.63425896942195, 14.13424662007286, 14.60798842878243, 15.24104587786780, 15.83090132315958, 16.12429654313971

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