Properties

Label 1-2205-2205.1004-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  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)8-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.988 + 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)32-s + (−0.0747 + 0.997i)34-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)8-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.988 + 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)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: $\chi_{2205} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395663220 + 1.010973692i\)
\(L(\frac12)\) \(\approx\) \(1.395663220 + 1.010973692i\)
\(L(1)\) \(\approx\) \(0.9998213431 + 0.5431100516i\)
\(L(1)\) \(\approx\) \(0.9998213431 + 0.5431100516i\)

Euler product

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

−19.635019349659437616281247697, −18.85036945592432933285186728056, −18.47544071020260651936800800205, −17.47177800181270790995087847228, −16.93619217186004683918331633688, −16.06753494441478164931594891479, −15.00167887006581989649851996567, −14.29461226233901944710771368827, −13.82391804663009370161671469535, −12.79603081702806181265468552932, −12.254572716672975944576442411932, −11.655159556630719229866603639603, −10.64375126306189767345635880319, −10.267374827649325352481919445479, −9.33150040745360652701154765124, −8.62302758503790257402367146353, −7.92428432308486092719395335976, −6.85586006780577828900582791223, −5.72999372053577477958611401086, −5.22179290039044458476277429947, −4.14602771850978218049406042062, −3.33459607943386919467021502991, −2.826293930327865485427236216831, −1.4672331172224865238477103963, −0.90901114174925813177528312592, 0.8096222150719405088726697640, 1.90740821976902021616004240979, 3.29859417162021997929049077325, 4.0385373643387936459189842683, 4.81301263974380950124434678408, 5.64701056360390008615886763192, 6.483956028728621322476978808286, 7.12350964854349882338133870241, 7.80790154481730257147379942919, 8.749769988807585513927117010825, 9.50244170407626878310674648001, 9.85769554172296542013347182846, 11.28275436938729861849520111212, 11.86017519195809620435941221788, 12.765711903706157297914840405572, 13.57919571970110137962530993211, 14.14808236450314650757623719533, 14.90306471933919562467549476150, 15.4620503279927281602776333779, 16.36931571386572589823989898945, 16.94248893563943293787743994859, 17.4951565940478996035111366225, 18.303096064301039800016780432923, 19.14343649254582296217669715690, 19.5559984739115490544934372080

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