Properties

Label 2-110-55.52-c1-0-2
Degree $2$
Conductor $110$
Sign $0.586 + 0.809i$
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  + (0.156 − 0.987i)2-s + (0.346 + 0.176i)3-s + (−0.951 − 0.309i)4-s + (2.14 − 0.617i)5-s + (0.228 − 0.314i)6-s + (−0.00697 − 0.0136i)7-s + (−0.453 + 0.891i)8-s + (−1.67 − 2.30i)9-s + (−0.273 − 2.21i)10-s + (2.43 + 2.25i)11-s + (−0.274 − 0.274i)12-s + (−2.31 − 0.366i)13-s + (−0.0146 + 0.00474i)14-s + (0.853 + 0.165i)15-s + (0.809 + 0.587i)16-s + (−3.88 + 0.615i)17-s + ⋯
L(s)  = 1  + (0.110 − 0.698i)2-s + (0.199 + 0.101i)3-s + (−0.475 − 0.154i)4-s + (0.961 − 0.276i)5-s + (0.0932 − 0.128i)6-s + (−0.00263 − 0.00517i)7-s + (−0.160 + 0.315i)8-s + (−0.558 − 0.768i)9-s + (−0.0865 − 0.701i)10-s + (0.733 + 0.679i)11-s + (−0.0793 − 0.0793i)12-s + (−0.641 − 0.101i)13-s + (−0.00390 + 0.00126i)14-s + (0.220 + 0.0426i)15-s + (0.202 + 0.146i)16-s + (−0.942 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.586 + 0.809i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.586 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06410 - 0.542977i\)
\(L(\frac12)\) \(\approx\) \(1.06410 - 0.542977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 + (-2.14 + 0.617i)T \)
11 \( 1 + (-2.43 - 2.25i)T \)
good3 \( 1 + (-0.346 - 0.176i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (0.00697 + 0.0136i)T + (-4.11 + 5.66i)T^{2} \)
13 \( 1 + (2.31 + 0.366i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (3.88 - 0.615i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.78 - 5.49i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.75 - 4.75i)T - 23iT^{2} \)
29 \( 1 + (1.28 - 3.94i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.70 + 3.41i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.00 - 0.512i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (4.46 - 1.45i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.68 + 5.68i)T + 43iT^{2} \)
47 \( 1 + (0.703 - 1.37i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (-2.19 + 13.8i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.86 - 0.606i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.84 + 6.66i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \)
71 \( 1 + (5.20 + 3.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.10 + 2.09i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-3.84 + 2.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.08 - 13.1i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 - 9.48iT - 89T^{2} \)
97 \( 1 + (-4.39 - 0.696i)T + (92.2 + 29.9i)T^{2} \)
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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

−13.50599440908379472472809676153, −12.35538459209133186631496713962, −11.65005859291185149352564328783, −10.05672850184484909264253085020, −9.552408528203841824089339731306, −8.431875270957042320774307263600, −6.58843265317878283087294013150, −5.31549992240307450069499224055, −3.74582799221677722891776370573, −1.96537329385867481936716967372, 2.60954933327480020273853161729, 4.72588256516286289339463174110, 6.01219482823237665179382192849, 6.99550780256590616510988321729, 8.434542708684422665916674977456, 9.316782569168442555607818531432, 10.57219200987611941703060027812, 11.78030463715615268885735068128, 13.31331224576152545063095277144, 13.85806267024178991079250386311

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