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Properties

Label 1-2205-2205.47-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.764 - 0.644i$
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  + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (−0.974 − 0.222i)8-s + (0.900 − 0.433i)11-s + (0.997 + 0.0747i)13-s + (0.955 + 0.294i)16-s + (−0.149 − 0.988i)17-s + (−0.5 − 0.866i)19-s + (−0.930 + 0.365i)22-s + (0.781 + 0.623i)23-s + (−0.988 − 0.149i)26-s + (0.365 − 0.930i)29-s + (0.5 + 0.866i)31-s + (−0.930 − 0.365i)32-s + (0.0747 + 0.997i)34-s + ⋯
L(s)  = 1  + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (−0.974 − 0.222i)8-s + (0.900 − 0.433i)11-s + (0.997 + 0.0747i)13-s + (0.955 + 0.294i)16-s + (−0.149 − 0.988i)17-s + (−0.5 − 0.866i)19-s + (−0.930 + 0.365i)22-s + (0.781 + 0.623i)23-s + (−0.988 − 0.149i)26-s + (0.365 − 0.930i)29-s + (0.5 + 0.866i)31-s + (−0.930 − 0.365i)32-s + (0.0747 + 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703761982 - 0.6221274872i\)
\(L(\frac12)\) \(\approx\) \(1.703761982 - 0.6221274872i\)
\(L(1)\) \(\approx\) \(0.8586115822 - 0.1103545298i\)
\(L(1)\) \(\approx\) \(0.8586115822 - 0.1103545298i\)

Euler product

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

−19.35787037276108198501985931335, −19.01275954576947485387738989504, −18.1407827177162565090783612490, −17.52353416170650354534272388573, −16.73830044432940086063053038768, −16.35578928144636110124143376991, −15.23195988047903046426840076804, −14.89007383384973794855637224913, −13.99111477958240286230262854803, −12.84245127261234416849148433120, −12.27875256563180109645336922558, −11.34291608059248062403457269861, −10.74140513333822629411428823762, −10.04734354278742381718935778918, −9.1812998738355949077178941344, −8.53786693374126588692899349438, −7.945803226062392053030603393280, −6.86177217790346440370845925810, −6.3618682658524475961934830936, −5.600805312992585608568918324671, −4.26233065550334232440548858013, −3.52265538782382421942302044537, −2.38200165536659010511991762902, −1.51381857716353421404934914111, −0.7787741813253625171068538352, 0.635588128306253329207872464806, 1.188749540537938199347462351840, 2.35761449413036147928918785198, 3.16179993222731316802585572178, 4.06920258146559630252886696433, 5.216979643196511173560070682244, 6.32095100741906086831140887570, 6.70718896665806756557037674496, 7.646896853576430151959288394017, 8.53442701447632820694974688501, 9.0758442449807227578023895270, 9.702943991730142154193412703836, 10.71766211967012766119178876311, 11.354911218831381974273737610, 11.78255464807044333617133217027, 12.856103036148973890370038570071, 13.67584146734693871018736869018, 14.45901021186705084007317637213, 15.541607233366319316056687810431, 15.830185646930559810800557347172, 16.75812347373094377861108977152, 17.39824943478101094252184772163, 17.94255613667245963836362848826, 18.96978879752996070101379684812, 19.18024985650864154258410777427

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