Properties

Label 2-110-1.1-c1-0-4
Degree $2$
Conductor $110$
Sign $1$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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 − 5-s + 6-s − 7-s + 8-s − 2·9-s − 10-s − 11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s − 3·17-s − 2·18-s − 19-s − 20-s − 21-s − 22-s + 6·23-s + 24-s + 25-s + 2·26-s − 5·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.540747344\)
\(L(\frac12)\) \(\approx\) \(1.540747344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.54712946528400016991259874830, −12.94269427172040951553295571447, −11.61430649233435892606346187577, −10.81523504438992298376035875160, −9.240245216843454718321617703983, −8.197325772660852886000765577503, −6.92339690043320565435471363407, −5.57585253794403166430858921429, −3.99189905466957435614609953053, −2.72390738347756879244011461463, 2.72390738347756879244011461463, 3.99189905466957435614609953053, 5.57585253794403166430858921429, 6.92339690043320565435471363407, 8.197325772660852886000765577503, 9.240245216843454718321617703983, 10.81523504438992298376035875160, 11.61430649233435892606346187577, 12.94269427172040951553295571447, 13.54712946528400016991259874830

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