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Properties

Label 1-2205-2205.1348-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.171 + 0.985i$
Analytic cond. $236.960$
Root an. cond. $236.960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.974 − 0.222i)8-s + (0.826 − 0.563i)11-s + (0.563 + 0.826i)13-s + (−0.222 + 0.974i)16-s + (−0.149 + 0.988i)17-s + (0.5 − 0.866i)19-s + (0.149 + 0.988i)22-s + (0.930 + 0.365i)23-s + (−0.988 + 0.149i)26-s + (0.988 + 0.149i)29-s + 31-s + (−0.781 − 0.623i)32-s + (−0.826 − 0.563i)34-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.974 − 0.222i)8-s + (0.826 − 0.563i)11-s + (0.563 + 0.826i)13-s + (−0.222 + 0.974i)16-s + (−0.149 + 0.988i)17-s + (0.5 − 0.866i)19-s + (0.149 + 0.988i)22-s + (0.930 + 0.365i)23-s + (−0.988 + 0.149i)26-s + (0.988 + 0.149i)29-s + 31-s + (−0.781 − 0.623i)32-s + (−0.826 − 0.563i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.171 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722996193 + 1.448823904i\)
\(L(\frac12)\) \(\approx\) \(1.722996193 + 1.448823904i\)
\(L(1)\) \(\approx\) \(0.9409548281 + 0.4231877493i\)
\(L(1)\) \(\approx\) \(0.9409548281 + 0.4231877493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.563 + 0.826i)T \)
17 \( 1 + (-0.149 + 0.988i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.930 + 0.365i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + T \)
37 \( 1 + (0.930 - 0.365i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.294 + 0.955i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.930 + 0.365i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.563 + 0.826i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.563 + 0.826i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.47888403453884530592606637653, −18.68170867613989690681127018179, −18.01808131589161393955245694970, −17.48021700492791250875306023139, −16.65706955153997629179940977082, −15.96906161292451200781551625481, −14.99160471802397199291936245343, −14.13882199471409335612775156068, −13.43655390447283634538666803656, −12.709138915717812661902453238421, −11.87167122515096894729620143935, −11.50352276783027481737084970146, −10.38392201130562129680188665906, −10.00673132779575687814158780957, −9.06218388307559993634003938574, −8.49148076852716575649732021071, −7.576502494584272436564921628340, −6.85598384122653134068150059529, −5.72691975815341239232450048056, −4.74333707070896949677232261747, −4.01685144877193253646466430352, −3.098165543058767905157507779401, −2.403917455794236808116365410498, −1.20891094371505180868744754033, −0.670112595224903869344158617027, 0.84175783920184471313048159763, 1.37844526639336158831410337967, 2.763475278559118999531049605639, 3.96410963637153378444827009053, 4.57174812546651188748643694359, 5.59707259071727141274180267973, 6.43763279383523128813033503652, 6.78761533348184054400081163543, 7.89086183268267167740681350144, 8.5689307736440237235281604896, 9.23557052038980057846419900815, 9.83365344922486664855558129637, 11.04822180078659153633234021787, 11.307538593656293109576062097780, 12.600466411862328679539894110490, 13.409181869351648492891855900216, 14.08541266796328015467064393723, 14.658143485922311144220066113465, 15.548071625555771710658967394012, 16.11310583575846775448534373401, 16.84618304228103154263410332956, 17.479614101522634006711790523868, 18.079419412365924023174555501665, 19.16007614813677897992257166958, 19.31574237510708712619322229741

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