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Properties

Label 2-2205-21.20-c1-0-2
Degree $2$
Conductor $2205$
Sign $0.860 + 0.508i$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
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  + 2.65i·2-s − 5.04·4-s − 5-s − 8.08i·8-s − 2.65i·10-s + 4.28i·11-s + 3.72i·13-s + 11.3·16-s − 5.64·17-s + 6.24i·19-s + 5.04·20-s − 11.3·22-s + 2.66i·23-s + 25-s − 9.87·26-s + ⋯
L(s)  = 1  + 1.87i·2-s − 2.52·4-s − 0.447·5-s − 2.85i·8-s − 0.839i·10-s + 1.29i·11-s + 1.03i·13-s + 2.84·16-s − 1.36·17-s + 1.43i·19-s + 1.12·20-s − 2.42·22-s + 0.556i·23-s + 0.200·25-s − 1.93·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2883742766\)
\(L(\frac12)\) \(\approx\) \(0.2883742766\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - 2.65iT - 2T^{2} \)
11 \( 1 - 4.28iT - 11T^{2} \)
13 \( 1 - 3.72iT - 13T^{2} \)
17 \( 1 + 5.64T + 17T^{2} \)
19 \( 1 - 6.24iT - 19T^{2} \)
23 \( 1 - 2.66iT - 23T^{2} \)
29 \( 1 + 10.7iT - 29T^{2} \)
31 \( 1 - 6.21iT - 31T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 + 4.50T + 41T^{2} \)
43 \( 1 - 4.89T + 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 + 9.01iT - 53T^{2} \)
59 \( 1 - 7.79T + 59T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 - 0.434T + 67T^{2} \)
71 \( 1 - 6.48iT - 71T^{2} \)
73 \( 1 + 7.40iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 8.90T + 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 - 3.04iT - 97T^{2} \)
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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

−9.635797481456148999530265795171, −8.615103080254473986376588777430, −8.101192087742069108748238445714, −7.29308069523883858732853241556, −6.74320861796971671267658466551, −6.11504420685060871080440163965, −5.05256966309982419387915971071, −4.39649918983589734367711678525, −3.79645809496703238071631302505, −1.90256940300889754161669107360, 0.11928028119005752631140259181, 1.05944792100274133867830212578, 2.54119918596584863427191181780, 3.05256325046368886346413173381, 3.98425264157909187209226178309, 4.77455452110702973438923739904, 5.59505290395816920677964114803, 6.80722353860485567517777446973, 7.992337924528828063234354327624, 8.848128943512438157246376985372

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