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Properties

Label 1-2205-2205.824-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.311 + 0.950i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)8-s + (−0.0747 − 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.365 + 0.930i)22-s + (−0.988 + 0.149i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s − 31-s + (0.623 − 0.781i)32-s + (−0.0747 + 0.997i)34-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)8-s + (−0.0747 − 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.365 + 0.930i)22-s + (−0.988 + 0.149i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s − 31-s + (0.623 − 0.781i)32-s + (−0.0747 + 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4849523849 + 0.3512839601i\)
\(L(\frac12)\) \(\approx\) \(0.4849523849 + 0.3512839601i\)
\(L(1)\) \(\approx\) \(0.6362997918 - 0.04739155573i\)
\(L(1)\) \(\approx\) \(0.6362997918 - 0.04739155573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 - T \)
37 \( 1 + (0.988 + 0.149i)T \)
41 \( 1 + (-0.733 + 0.680i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−19.65405386050181895700001491651, −18.67271220708292801241872305021, −18.015288175660241616910701035532, −17.53819465449536791851788621421, −16.87351425610902865355923843816, −15.90860105524602787535726385152, −15.41087013074131560088429204973, −14.78603081732192712907596177559, −13.99594404676687353383482461548, −12.91445162252963125979795960416, −12.31937970503151204187647815334, −11.25897286817582377645952349790, −10.612280270933003780303254795235, −9.97593658684228374568475842960, −9.1816059486056053388711512546, −8.481974640383016189786149917190, −7.59600970698817337769426406485, −7.1338069974782846792336151804, −6.1367546474106657432253002578, −5.448519853118559940617234722550, −4.56768070613535148264520153304, −3.39813943165881394996630924398, −2.29102155742996506955410294338, −1.59018027160772426560558737048, −0.30313351677837776331244518159, 0.99481462737583534849364955283, 1.93193822025564503465786680555, 2.81614151260817428490379122959, 3.68051517622128099687675315282, 4.485355674266025721282492434779, 5.85514150302275860168409392459, 6.403423693308774903965214327719, 7.551385468420382983561719393225, 7.920089969192737937419363045399, 9.05502540106544288719851669202, 9.36701033098067739337873321463, 10.294042748333912954934429831034, 11.12681886606671647780289208643, 11.66801655937847841193094225419, 12.27323217658375672637819530261, 13.3984503332963162858430498835, 13.88507105277343090370726341308, 14.90079294394187322506392467858, 15.9456991630003008530859074404, 16.39844711811184894698505621507, 16.88404608052747085705962067200, 18.06231528792031336565191513910, 18.34288083509361930786857036535, 19.1361065887430451689586115303, 19.72151430468697401591473898834

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