Properties

Label 1-2205-2205.1769-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.682 + 0.730i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)8-s + (−0.955 + 0.294i)11-s + (0.955 − 0.294i)13-s + (0.623 + 0.781i)16-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)22-s + (0.826 − 0.563i)23-s + (0.0747 + 0.997i)26-s + (−0.0747 + 0.997i)29-s − 31-s + (−0.900 + 0.433i)32-s + (−0.955 − 0.294i)34-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)8-s + (−0.955 + 0.294i)11-s + (0.955 − 0.294i)13-s + (0.623 + 0.781i)16-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)22-s + (0.826 − 0.563i)23-s + (0.0747 + 0.997i)26-s + (−0.0747 + 0.997i)29-s − 31-s + (−0.900 + 0.433i)32-s + (−0.955 − 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.682 + 0.730i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.682 + 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4236452848 + 0.9757427619i\)
\(L(\frac12)\) \(\approx\) \(0.4236452848 + 0.9757427619i\)
\(L(1)\) \(\approx\) \(0.7264138550 + 0.4458983851i\)
\(L(1)\) \(\approx\) \(0.7264138550 + 0.4458983851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (-0.0747 + 0.997i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.40249081279678629665263461813, −18.77658260115463446075400807116, −18.192050188731756218537255370445, −17.58283716899700910602823898616, −16.68228714744607253823552289386, −15.901995903341124849160825652982, −15.22314933937274669944482244908, −13.95644049885804461243270860596, −13.56336832555100110728125640856, −12.96475560492212423459783454186, −12.01971055393428319867621700195, −11.20191339769311043382630493046, −10.924443824528206001470606945798, −9.874854490568285433260634133766, −9.22377870587821262148927011315, −8.541595803734065900243972132783, −7.68972602023639313348355165762, −6.91339863113218105120375626601, −5.59724300295110669976545604806, −5.02252247715038879749489335644, −4.05832825180811007707723623584, −3.15342166085568956818138969495, −2.55183151849333152838095008107, −1.466224672476565396918816280643, −0.475192896677169112213663438130, 0.98701270010624627310333046412, 2.012736060261931043781385177475, 3.4277878575850730409110801367, 4.047449472458793169238154969262, 5.380877957451158770324827774177, 5.47366563671457669057218242748, 6.655325419295710249324298477513, 7.2591223265336475911519774259, 8.2127199214292093100288964584, 8.60964423401639818363619171160, 9.53576275637093655020697136437, 10.59045598144358109059624306946, 10.70478605112465257363864094548, 12.2282743685671315830576734343, 12.938340678210593373278736398428, 13.48307563228109221183874811158, 14.426574831589910423283516578839, 15.00659354089239260719238960333, 15.7396011463245653478780249409, 16.34217820070383392853681710008, 17.01419780200951397895160197223, 17.90936930409250774824863356873, 18.3827264699359115230007357192, 18.97421643142359999155469772920, 19.92376528668131423555903835030

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