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Properties

Label 2-110-55.32-c1-0-3
Degree $2$
Conductor $110$
Sign $0.993 + 0.112i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.70 − 1.70i)3-s − 1.00i·4-s + (−0.707 + 2.12i)5-s + 2.41i·6-s + (2.12 − 2.12i)7-s + (0.707 + 0.707i)8-s − 2.82i·9-s + (−0.999 − 2i)10-s + (−1.41 − 3i)11-s + (−1.70 − 1.70i)12-s + (3 + 3i)13-s + 3i·14-s + (2.41 + 4.82i)15-s − 1.00·16-s + (−5.12 + 5.12i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.985 − 0.985i)3-s − 0.500i·4-s + (−0.316 + 0.948i)5-s + 0.985i·6-s + (0.801 − 0.801i)7-s + (0.250 + 0.250i)8-s − 0.942i·9-s + (−0.316 − 0.632i)10-s + (−0.426 − 0.904i)11-s + (−0.492 − 0.492i)12-s + (0.832 + 0.832i)13-s + 0.801i·14-s + (0.623 + 1.24i)15-s − 0.250·16-s + (−1.24 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06431 - 0.0602316i\)
\(L(\frac12)\) \(\approx\) \(1.06431 - 0.0602316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 2.12i)T \)
11 \( 1 + (1.41 + 3i)T \)
good3 \( 1 + (-1.70 + 1.70i)T - 3iT^{2} \)
7 \( 1 + (-2.12 + 2.12i)T - 7iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (5.12 - 5.12i)T - 17iT^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (0.171 - 0.171i)T - 23iT^{2} \)
29 \( 1 + 1.24T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + (-0.121 - 0.121i)T + 37iT^{2} \)
41 \( 1 + 1.75iT - 41T^{2} \)
43 \( 1 + (-1.24 - 1.24i)T + 43iT^{2} \)
47 \( 1 + (-4.41 - 4.41i)T + 47iT^{2} \)
53 \( 1 + (-9.53 + 9.53i)T - 53iT^{2} \)
59 \( 1 - 1.41iT - 59T^{2} \)
61 \( 1 + 7.24iT - 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (7.24 + 7.24i)T + 83iT^{2} \)
89 \( 1 - 5.48iT - 89T^{2} \)
97 \( 1 + (2.24 + 2.24i)T + 97iT^{2} \)
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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

−13.87683186895264121145081919803, −13.05302755654893375889678190122, −11.16676462176742677992596977828, −10.70792309637563619273071001122, −8.808774128457611489355282898851, −8.129987811182682202157461426846, −7.20494011903028957006323110608, −6.28439109269128197840656246327, −3.89226970392756088836233050041, −1.96414213407946840078860083951, 2.35787865031481513039117495277, 4.07323826733133829954506404219, 5.14466741477393392422053021911, 7.67611878095049031690744826134, 8.779940724551212268075874414571, 9.059519036598965859776633533989, 10.36981264734809617538451475191, 11.46179716808661664722869591683, 12.59506143094397254949578924037, 13.59086556551898475667260223841

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