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Properties

Label 1-2205-2205.922-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.972 + 0.234i$
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  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (−0.997 + 0.0747i)13-s + (−0.222 + 0.974i)16-s + (0.930 − 0.365i)17-s + (−0.5 − 0.866i)19-s + (0.930 + 0.365i)22-s + (0.149 + 0.988i)23-s + (−0.365 + 0.930i)26-s + (−0.365 − 0.930i)29-s − 31-s + (0.781 + 0.623i)32-s + (0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (−0.997 + 0.0747i)13-s + (−0.222 + 0.974i)16-s + (0.930 − 0.365i)17-s + (−0.5 − 0.866i)19-s + (0.930 + 0.365i)22-s + (0.149 + 0.988i)23-s + (−0.365 + 0.930i)26-s + (−0.365 − 0.930i)29-s − 31-s + (0.781 + 0.623i)32-s + (0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (922, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.972 + 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149427517 + 0.1365951122i\)
\(L(\frac12)\) \(\approx\) \(1.149427517 + 0.1365951122i\)
\(L(1)\) \(\approx\) \(0.9760982256 - 0.3798705209i\)
\(L(1)\) \(\approx\) \(0.9760982256 - 0.3798705209i\)

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 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
17 \( 1 + (0.930 - 0.365i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.149 + 0.988i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 - T \)
37 \( 1 + (0.149 - 0.988i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.680 + 0.733i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.149 + 0.988i)T \)
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 \( 1 + (0.997 + 0.0747i)T \)
79 \( 1 - T \)
83 \( 1 + (0.997 + 0.0747i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−19.658331234369429319051864633348, −18.65538397250382179135083588235, −18.40084004349729176024858699289, −17.19046995522634639126154952642, −16.6601272874227160505601134871, −16.34709595097302485874427171946, −15.20791889526813346695973668876, −14.58103961615289042746746369250, −14.20678478041033628924160818216, −13.19316062086407326899070458244, −12.54939384383962748582769282124, −11.95883528273325776055479202545, −10.91328575824054038762947252071, −10.028004128855414889979849002093, −9.18917224121471813946440122875, −8.301735412576828135033077018764, −7.85999920078726756577728903716, −6.865381746460684192765381575492, −6.20146504538582465635525234720, −5.383072742924155124251037830258, −4.75855399603986415896687173238, −3.63576957223085324040749700048, −3.1561398850093543405311603565, −1.8783089454917046548719122114, −0.36539589647998118274337282757, 1.02977801611699598199020443412, 2.090782280824455661048778616759, 2.684139250354521746422184381779, 3.738643797615833369332359665259, 4.50622556998933389088506653504, 5.211142102460411628927658681856, 5.97871223580228922858489721175, 7.150425988145694075053840863804, 7.723960787705790603971030315257, 9.10323779251291128973577540672, 9.50205213415617369833551147268, 10.21422045875491349673288088587, 11.07350813558370560289977295892, 11.791949601039485177076159925921, 12.455237903862912702559845803101, 13.03596524660171100307642096654, 13.87304030303768330783708213696, 14.717525167559216764672373322098, 15.06245955429899851768707194347, 16.04271769022686247165404897912, 17.17326945267599441174017394797, 17.638727994477592212831858957299, 18.46423238438528366568780323761, 19.29188489240590255191745149647, 19.775023571433570908398307668399

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