Properties

Degree $2$
Conductor $110$
Sign $1$
Motivic weight $1$
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 + 5·7-s − 8-s − 2·9-s + 10-s + 11-s + 12-s + 2·13-s − 5·14-s − 15-s + 16-s + 3·17-s + 2·18-s − 7·19-s − 20-s + 5·21-s − 22-s − 6·23-s − 24-s + 25-s − 2·26-s − 5·27-s + 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.88·7-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 − 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.60·19-s − 0.223·20-s + 1.09·21-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 + 0.944·28-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: $1$
Motivic weight: \(1\)
Character: $\chi_{110} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9420579665\)
\(L(\frac12)\) \(\approx\) \(0.9420579665\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−19.32704181196344, −18.46000402182116, −17.52251662599698, −16.84579337197409, −15.52740236396597, −14.57255006605518, −14.13397147037926, −12.37150177127700, −11.32146429554651, −10.70422776242305, −8.996361573480265, −8.285662434899333, −7.569811964690275, −5.738764390858290, −4.011731357475171, −1.971708425879564, 1.971708425879564, 4.011731357475171, 5.738764390858290, 7.569811964690275, 8.285662434899333, 8.996361573480265, 10.70422776242305, 11.32146429554651, 12.37150177127700, 14.13397147037926, 14.57255006605518, 15.52740236396597, 16.84579337197409, 17.52251662599698, 18.46000402182116, 19.32704181196344

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