↑

Properties

Label 1-2205-2205.2182-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.285 - 0.958i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (0.149 − 0.988i)13-s + (−0.900 + 0.433i)16-s + (−0.680 + 0.733i)17-s + (−0.5 − 0.866i)19-s + (−0.680 − 0.733i)22-s + (0.294 − 0.955i)23-s + (0.733 − 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.974 − 0.222i)32-s + (−0.988 + 0.149i)34-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (0.149 − 0.988i)13-s + (−0.900 + 0.433i)16-s + (−0.680 + 0.733i)17-s + (−0.5 − 0.866i)19-s + (−0.680 − 0.733i)22-s + (0.294 − 0.955i)23-s + (0.733 − 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.974 − 0.222i)32-s + (−0.988 + 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.285 - 0.958i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.285 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6866561540 - 0.5116738752i\)
\(L(\frac12)\) \(\approx\) \(0.6866561540 - 0.5116738752i\)
\(L(1)\) \(\approx\) \(1.147214947 + 0.3366354518i\)
\(L(1)\) \(\approx\) \(1.147214947 + 0.3366354518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (0.149 - 0.988i)T \)
17 \( 1 + (-0.680 + 0.733i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.294 - 0.955i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (0.294 + 0.955i)T \)
41 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (-0.997 - 0.0747i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.294 - 0.955i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.149 - 0.988i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.149 - 0.988i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.954735530446642036785393139934, −19.32062542662093218586097546100, −18.47609686735492752891389541062, −18.0779776878621620919630635978, −16.8815345329848840461054917773, −16.037422492700765664318550867517, −15.50649655941708221953466867920, −14.645503773566832231197170928129, −13.98283937520001331816352598087, −13.25550189989317892130111743096, −12.72582233193358323858253028307, −11.76482380048253968546211749944, −11.2629814016099314938039759359, −10.47885770257855937610231236788, −9.69301082511716330525854896006, −9.03118462860917888034144862871, −7.90513328756480227057591414746, −7.01481525074640713574899965285, −6.219179049410603750972304095409, −5.40163093178807296878659626978, −4.62280392115128920637509510070, −3.93404607564357148629341219404, −2.94054329124088096497973026822, −2.17978833389926953662184507327, −1.3410311987925311135427606262, 0.19628071063387000122146120937, 1.95288405586831383119517183438, 2.85050492009667874980669818137, 3.514278564955638285281630924624, 4.6587453436074171631452675853, 5.112155432764567476210610955043, 6.05513602037769964366250145939, 6.73345558803488566738302750352, 7.55173374935991551542192283651, 8.44037219816931844571208376370, 8.77756275973781808126984614049, 10.27324016599409801284982542924, 10.789316080645891502492014756524, 11.64245785475538514260889070102, 12.73685517061567332992843237746, 12.98269976634372854189191367826, 13.673254817872223642779189819475, 14.727191262267555432863124056298, 15.17738351280171892048357660392, 15.84033874125448295345124889354, 16.544806762466250785612552398972, 17.37574029878025820087083028140, 17.961290009326947492285970398448, 18.68983906200171930689779951954, 19.88774717838865657870692165708

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